How do you verify the identity #1/(1-sintheta)+1/(1+sintheta)=2sec^2theta#?

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To verify the identity ( \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} = 2 \sec^2(\theta) ), we can start with the left-hand side of the equation and simplify it step by step.

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StartingTo verify the identity ( \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} = 2 \sec^2(\theta) ), we can start with the left-hand side of the equation and simplify it step by step.

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Starting withTo verify the identity ( \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} = 2 \sec^2(\theta) ), we can start with the left-hand side of the equation and simplify it step by step.

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Starting with LTo verify the identity ( \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} = 2 \sec^2(\theta) ), we can start with the left-hand side of the equation and simplify it step by step.

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Starting with LHSTo verify the identity ( \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} = 2 \sec^2(\theta) ), we can start with the left-hand side of the equation and simplify it step by step.

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Starting with LHS: To verify the identity ( \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} = 2 \sec^2(\theta) ), we can start with the left-hand side of the equation and simplify it step by step.

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Starting with LHS: [To verify the identity ( \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} = 2 \sec^2(\theta) ), we can start with the left-hand side of the equation and simplify it step by step.

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Starting with LHS: [ \To verify the identity ( \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} = 2 \sec^2(\theta) ), we can start with the left-hand side of the equation and simplify it step by step.

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Starting with LHS: [ \frac{1To verify the identity ( \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} = 2 \sec^2(\theta) ), we can start with the left-hand side of the equation and simplify it step by step.

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Starting with LHS: [ \frac{1}{To verify the identity ( \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} = 2 \sec^2(\theta) ), we can start with the left-hand side of the equation and simplify it step by step.

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Starting with LHS: [ \frac{1}{1-\sin\theta} + \frac{1}{To verify the identity ( \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} = 2 \sec^2(\theta) ), we can start with the left-hand side of the equation and simplify it step by step.

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Starting with LHS: [ \frac{1}{1-\sin\theta} + \frac{1}{1To verify the identity ( \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} = 2 \sec^2(\theta) ), we can start with the left-hand side of the equation and simplify it step by step.

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Starting with LHS: [ \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\To verify the identity ( \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} = 2 \sec^2(\theta) ), we can start with the left-hand side of the equation and simplify it step by step.

[ \frac{1}{1 - \sin(\theta)} + \frac{1To verify the identity ( \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} = 2\sec^2\theta ), we can start with the left-hand side (LHS) of the equation and manipulate it until it matches the right-hand side (RHS).

Starting with LHS: [ \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta}To verify the identity ( \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} = 2 \sec^2(\theta) ), we can start with the left-hand side of the equation and simplify it step by step.

[ \frac{1}{1 - \sin(\theta)} + \frac{1}{To verify the identity ( \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} = 2\sec^2\theta ), we can start with the left-hand side (LHS) of the equation and manipulate it until it matches the right-hand side (RHS).

Starting with LHS: [ \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} ]

To verify the identity ( \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} = 2 \sec^2(\theta) ), we can start with the left-hand side of the equation and simplify it step by step.

[ \frac{1}{1 - \sin(\theta)} + \frac{1}{1To verify the identity ( \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} = 2\sec^2\theta ), we can start with the left-hand side (LHS) of the equation and manipulate it until it matches the right-hand side (RHS).

Starting with LHS: [ \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} ]

To combine theseTo verify the identity ( \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} = 2 \sec^2(\theta) ), we can start with the left-hand side of the equation and simplify it step by step.

[ \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \To verify the identity ( \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} = 2\sec^2\theta ), we can start with the left-hand side (LHS) of the equation and manipulate it until it matches the right-hand side (RHS).

Starting with LHS: [ \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} ]

To combine these fractionsTo verify the identity ( \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} = 2 \sec^2(\theta) ), we can start with the left-hand side of the equation and simplify it step by step.

[ \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} ]

To combine theseTo verify the identity ( \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} = 2\sec^2\theta ), we can start with the left-hand side (LHS) of the equation and manipulate it until it matches the right-hand side (RHS).

Starting with LHS: [ \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} ]

To combine these fractions,To verify the identity ( \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} = 2 \sec^2(\theta) ), we can start with the left-hand side of the equation and simplify it step by step.

[ \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} ]

To combine these fractionsTo verify the identity ( \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} = 2\sec^2\theta ), we can start with the left-hand side (LHS) of the equation and manipulate it until it matches the right-hand side (RHS).

Starting with LHS: [ \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} ]

To combine these fractions, weTo verify the identity ( \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} = 2 \sec^2(\theta) ), we can start with the left-hand side of the equation and simplify it step by step.

[ \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} ]

To combine these fractions,To verify the identity ( \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} = 2\sec^2\theta ), we can start with the left-hand side (LHS) of the equation and manipulate it until it matches the right-hand side (RHS).

Starting with LHS: [ \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} ]

To combine these fractions, we needTo verify the identity ( \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} = 2 \sec^2(\theta) ), we can start with the left-hand side of the equation and simplify it step by step.

[ \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} ]

To combine these fractions, weTo verify the identity ( \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} = 2\sec^2\theta ), we can start with the left-hand side (LHS) of the equation and manipulate it until it matches the right-hand side (RHS).

Starting with LHS: [ \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} ]

To combine these fractions, we need a commonTo verify the identity ( \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} = 2 \sec^2(\theta) ), we can start with the left-hand side of the equation and simplify it step by step.

[ \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} ]

To combine these fractions, we findTo verify the identity ( \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} = 2\sec^2\theta ), we can start with the left-hand side (LHS) of the equation and manipulate it until it matches the right-hand side (RHS).

Starting with LHS: [ \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} ]

To combine these fractions, we need a common denominatorTo verify the identity ( \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} = 2 \sec^2(\theta) ), we can start with the left-hand side of the equation and simplify it step by step.

[ \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} ]

To combine these fractions, we find aTo verify the identity ( \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} = 2\sec^2\theta ), we can start with the left-hand side (LHS) of the equation and manipulate it until it matches the right-hand side (RHS).

Starting with LHS: [ \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} ]

To combine these fractions, we need a common denominator.To verify the identity ( \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} = 2 \sec^2(\theta) ), we can start with the left-hand side of the equation and simplify it step by step.

[ \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} ]

To combine these fractions, we find a commonTo verify the identity ( \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} = 2\sec^2\theta ), we can start with the left-hand side (LHS) of the equation and manipulate it until it matches the right-hand side (RHS).

Starting with LHS: [ \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} ]

To combine these fractions, we need a common denominator. TheTo verify the identity ( \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} = 2 \sec^2(\theta) ), we can start with the left-hand side of the equation and simplify it step by step.

[ \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} ]

To combine these fractions, we find a common denominatorTo verify the identity ( \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} = 2\sec^2\theta ), we can start with the left-hand side (LHS) of the equation and manipulate it until it matches the right-hand side (RHS).

Starting with LHS: [ \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} ]

To combine these fractions, we need a common denominator. The commonTo verify the identity ( \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} = 2 \sec^2(\theta) ), we can start with the left-hand side of the equation and simplify it step by step.

[ \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} ]

To combine these fractions, we find a common denominator:

To verify the identity ( \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} = 2\sec^2\theta ), we can start with the left-hand side (LHS) of the equation and manipulate it until it matches the right-hand side (RHS).

Starting with LHS: [ \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} ]

To combine these fractions, we need a common denominator. The common denominatorTo verify the identity ( \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} = 2 \sec^2(\theta) ), we can start with the left-hand side of the equation and simplify it step by step.

[ \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} ]

To combine these fractions, we find a common denominator:

[To verify the identity ( \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} = 2\sec^2\theta ), we can start with the left-hand side (LHS) of the equation and manipulate it until it matches the right-hand side (RHS).

Starting with LHS: [ \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} ]

To combine these fractions, we need a common denominator. The common denominator hereTo verify the identity ( \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} = 2 \sec^2(\theta) ), we can start with the left-hand side of the equation and simplify it step by step.

[ \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} ]

To combine these fractions, we find a common denominator:

[ =To verify the identity ( \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} = 2\sec^2\theta ), we can start with the left-hand side (LHS) of the equation and manipulate it until it matches the right-hand side (RHS).

Starting with LHS: [ \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} ]

To combine these fractions, we need a common denominator. The common denominator here isTo verify the identity ( \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} = 2 \sec^2(\theta) ), we can start with the left-hand side of the equation and simplify it step by step.

[ \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} ]

To combine these fractions, we find a common denominator:

[ = \To verify the identity ( \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} = 2\sec^2\theta ), we can start with the left-hand side (LHS) of the equation and manipulate it until it matches the right-hand side (RHS).

Starting with LHS: [ \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} ]

To combine these fractions, we need a common denominator. The common denominator here is \To verify the identity ( \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} = 2 \sec^2(\theta) ), we can start with the left-hand side of the equation and simplify it step by step.

[ \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} ]

To combine these fractions, we find a common denominator:

[ = \fracTo verify the identity ( \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} = 2\sec^2\theta ), we can start with the left-hand side (LHS) of the equation and manipulate it until it matches the right-hand side (RHS).

Starting with LHS: [ \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} ]

To combine these fractions, we need a common denominator. The common denominator here is ((To verify the identity ( \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} = 2 \sec^2(\theta) ), we can start with the left-hand side of the equation and simplify it step by step.

[ \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} ]

To combine these fractions, we find a common denominator:

[ = \frac{(To verify the identity ( \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} = 2\sec^2\theta ), we can start with the left-hand side (LHS) of the equation and manipulate it until it matches the right-hand side (RHS).

Starting with LHS: [ \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} ]

To combine these fractions, we need a common denominator. The common denominator here is ((1To verify the identity ( \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} = 2 \sec^2(\theta) ), we can start with the left-hand side of the equation and simplify it step by step.

[ \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} ]

To combine these fractions, we find a common denominator:

[ = \frac{(1To verify the identity ( \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} = 2\sec^2\theta ), we can start with the left-hand side (LHS) of the equation and manipulate it until it matches the right-hand side (RHS).

Starting with LHS: [ \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} ]

To combine these fractions, we need a common denominator. The common denominator here is ((1-\To verify the identity ( \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} = 2 \sec^2(\theta) ), we can start with the left-hand side of the equation and simplify it step by step.

[ \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} ]

To combine these fractions, we find a common denominator:

[ = \frac{(1 + \To verify the identity ( \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} = 2\sec^2\theta ), we can start with the left-hand side (LHS) of the equation and manipulate it until it matches the right-hand side (RHS).

Starting with LHS: [ \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} ]

To combine these fractions, we need a common denominator. The common denominator here is ((1-\sinTo verify the identity ( \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} = 2 \sec^2(\theta) ), we can start with the left-hand side of the equation and simplify it step by step.

[ \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} ]

To combine these fractions, we find a common denominator:

[ = \frac{(1 + \sinTo verify the identity ( \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} = 2\sec^2\theta ), we can start with the left-hand side (LHS) of the equation and manipulate it until it matches the right-hand side (RHS).

Starting with LHS: [ \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} ]

To combine these fractions, we need a common denominator. The common denominator here is ((1-\sin\To verify the identity ( \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} = 2 \sec^2(\theta) ), we can start with the left-hand side of the equation and simplify it step by step.

[ \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} ]

To combine these fractions, we find a common denominator:

[ = \frac{(1 + \sin(\To verify the identity ( \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} = 2\sec^2\theta ), we can start with the left-hand side (LHS) of the equation and manipulate it until it matches the right-hand side (RHS).

Starting with LHS: [ \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} ]

To combine these fractions, we need a common denominator. The common denominator here is ((1-\sin\thetaTo verify the identity ( \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} = 2 \sec^2(\theta) ), we can start with the left-hand side of the equation and simplify it step by step.

[ \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} ]

To combine these fractions, we find a common denominator:

[ = \frac{(1 + \sin(\thetaTo verify the identity ( \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} = 2\sec^2\theta ), we can start with the left-hand side (LHS) of the equation and manipulate it until it matches the right-hand side (RHS).

Starting with LHS: [ \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} ]

To combine these fractions, we need a common denominator. The common denominator here is ((1-\sin\theta)(To verify the identity ( \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} = 2 \sec^2(\theta) ), we can start with the left-hand side of the equation and simplify it step by step.

[ \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} ]

To combine these fractions, we find a common denominator:

[ = \frac{(1 + \sin(\theta))To verify the identity ( \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} = 2\sec^2\theta ), we can start with the left-hand side (LHS) of the equation and manipulate it until it matches the right-hand side (RHS).

Starting with LHS: [ \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} ]

To combine these fractions, we need a common denominator. The common denominator here is ((1-\sin\theta)(1To verify the identity ( \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} = 2 \sec^2(\theta) ), we can start with the left-hand side of the equation and simplify it step by step.

[ \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} ]

To combine these fractions, we find a common denominator:

[ = \frac{(1 + \sin(\theta)) +To verify the identity ( \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} = 2\sec^2\theta ), we can start with the left-hand side (LHS) of the equation and manipulate it until it matches the right-hand side (RHS).

Starting with LHS: [ \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} ]

To combine these fractions, we need a common denominator. The common denominator here is ((1-\sin\theta)(1+\To verify the identity ( \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} = 2 \sec^2(\theta) ), we can start with the left-hand side of the equation and simplify it step by step.

[ \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} ]

To combine these fractions, we find a common denominator:

[ = \frac{(1 + \sin(\theta)) + (To verify the identity ( \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} = 2\sec^2\theta ), we can start with the left-hand side (LHS) of the equation and manipulate it until it matches the right-hand side (RHS).

Starting with LHS: [ \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} ]

To combine these fractions, we need a common denominator. The common denominator here is ((1-\sin\theta)(1+\sinTo verify the identity ( \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} = 2 \sec^2(\theta) ), we can start with the left-hand side of the equation and simplify it step by step.

[ \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} ]

To combine these fractions, we find a common denominator:

[ = \frac{(1 + \sin(\theta)) + (1To verify the identity ( \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} = 2\sec^2\theta ), we can start with the left-hand side (LHS) of the equation and manipulate it until it matches the right-hand side (RHS).

Starting with LHS: [ \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} ]

To combine these fractions, we need a common denominator. The common denominator here is ((1-\sin\theta)(1+\sin\To verify the identity ( \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} = 2 \sec^2(\theta) ), we can start with the left-hand side of the equation and simplify it step by step.

[ \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} ]

To combine these fractions, we find a common denominator:

[ = \frac{(1 + \sin(\theta)) + (1 -To verify the identity ( \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} = 2\sec^2\theta ), we can start with the left-hand side (LHS) of the equation and manipulate it until it matches the right-hand side (RHS).

Starting with LHS: [ \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} ]

To combine these fractions, we need a common denominator. The common denominator here is ((1-\sin\theta)(1+\sin\thetaTo verify the identity ( \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} = 2 \sec^2(\theta) ), we can start with the left-hand side of the equation and simplify it step by step.

[ \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} ]

To combine these fractions, we find a common denominator:

[ = \frac{(1 + \sin(\theta)) + (1 - \To verify the identity ( \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} = 2\sec^2\theta ), we can start with the left-hand side (LHS) of the equation and manipulate it until it matches the right-hand side (RHS).

Starting with LHS: [ \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} ]

To combine these fractions, we need a common denominator. The common denominator here is ((1-\sin\theta)(1+\sin\theta)\To verify the identity ( \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} = 2 \sec^2(\theta) ), we can start with the left-hand side of the equation and simplify it step by step.

[ \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} ]

To combine these fractions, we find a common denominator:

[ = \frac{(1 + \sin(\theta)) + (1 - \sinTo verify the identity ( \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} = 2\sec^2\theta ), we can start with the left-hand side (LHS) of the equation and manipulate it until it matches the right-hand side (RHS).

Starting with LHS: [ \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} ]

To combine these fractions, we need a common denominator. The common denominator here is ((1-\sin\theta)(1+\sin\theta)).

To verify the identity ( \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} = 2 \sec^2(\theta) ), we can start with the left-hand side of the equation and simplify it step by step.

[ \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} ]

To combine these fractions, we find a common denominator:

[ = \frac{(1 + \sin(\theta)) + (1 - \sin(\thetaTo verify the identity ( \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} = 2\sec^2\theta ), we can start with the left-hand side (LHS) of the equation and manipulate it until it matches the right-hand side (RHS).

Starting with LHS: [ \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} ]

To combine these fractions, we need a common denominator. The common denominator here is ((1-\sin\theta)(1+\sin\theta)).

[To verify the identity ( \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} = 2 \sec^2(\theta) ), we can start with the left-hand side of the equation and simplify it step by step.

[ \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} ]

To combine these fractions, we find a common denominator:

[ = \frac{(1 + \sin(\theta)) + (1 - \sin(\theta))To verify the identity ( \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} = 2\sec^2\theta ), we can start with the left-hand side (LHS) of the equation and manipulate it until it matches the right-hand side (RHS).

Starting with LHS: [ \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} ]

To combine these fractions, we need a common denominator. The common denominator here is ((1-\sin\theta)(1+\sin\theta)).

[ \To verify the identity ( \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} = 2 \sec^2(\theta) ), we can start with the left-hand side of the equation and simplify it step by step.

[ \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} ]

To combine these fractions, we find a common denominator:

[ = \frac{(1 + \sin(\theta)) + (1 - \sin(\theta))}{To verify the identity ( \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} = 2\sec^2\theta ), we can start with the left-hand side (LHS) of the equation and manipulate it until it matches the right-hand side (RHS).

Starting with LHS: [ \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} ]

To combine these fractions, we need a common denominator. The common denominator here is ((1-\sin\theta)(1+\sin\theta)).

[ \fracTo verify the identity ( \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} = 2 \sec^2(\theta) ), we can start with the left-hand side of the equation and simplify it step by step.

[ \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} ]

To combine these fractions, we find a common denominator:

[ = \frac{(1 + \sin(\theta)) + (1 - \sin(\theta))}{(To verify the identity ( \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} = 2\sec^2\theta ), we can start with the left-hand side (LHS) of the equation and manipulate it until it matches the right-hand side (RHS).

Starting with LHS: [ \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} ]

To combine these fractions, we need a common denominator. The common denominator here is ((1-\sin\theta)(1+\sin\theta)).

[ \frac{(To verify the identity ( \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} = 2 \sec^2(\theta) ), we can start with the left-hand side of the equation and simplify it step by step.

[ \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} ]

To combine these fractions, we find a common denominator:

[ = \frac{(1 + \sin(\theta)) + (1 - \sin(\theta))}{(1To verify the identity ( \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} = 2\sec^2\theta ), we can start with the left-hand side (LHS) of the equation and manipulate it until it matches the right-hand side (RHS).

Starting with LHS: [ \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} ]

To combine these fractions, we need a common denominator. The common denominator here is ((1-\sin\theta)(1+\sin\theta)).

[ \frac{(1To verify the identity ( \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} = 2 \sec^2(\theta) ), we can start with the left-hand side of the equation and simplify it step by step.

[ \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} ]

To combine these fractions, we find a common denominator:

[ = \frac{(1 + \sin(\theta)) + (1 - \sin(\theta))}{(1 -To verify the identity ( \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} = 2\sec^2\theta ), we can start with the left-hand side (LHS) of the equation and manipulate it until it matches the right-hand side (RHS).

Starting with LHS: [ \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} ]

To combine these fractions, we need a common denominator. The common denominator here is ((1-\sin\theta)(1+\sin\theta)).

[ \frac{(1+\To verify the identity ( \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} = 2 \sec^2(\theta) ), we can start with the left-hand side of the equation and simplify it step by step.

[ \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} ]

To combine these fractions, we find a common denominator:

[ = \frac{(1 + \sin(\theta)) + (1 - \sin(\theta))}{(1 - \To verify the identity ( \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} = 2\sec^2\theta ), we can start with the left-hand side (LHS) of the equation and manipulate it until it matches the right-hand side (RHS).

Starting with LHS: [ \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} ]

To combine these fractions, we need a common denominator. The common denominator here is ((1-\sin\theta)(1+\sin\theta)).

[ \frac{(1+\sinTo verify the identity ( \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} = 2 \sec^2(\theta) ), we can start with the left-hand side of the equation and simplify it step by step.

[ \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} ]

To combine these fractions, we find a common denominator:

[ = \frac{(1 + \sin(\theta)) + (1 - \sin(\theta))}{(1 - \sinTo verify the identity ( \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} = 2\sec^2\theta ), we can start with the left-hand side (LHS) of the equation and manipulate it until it matches the right-hand side (RHS).

Starting with LHS: [ \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} ]

To combine these fractions, we need a common denominator. The common denominator here is ((1-\sin\theta)(1+\sin\theta)).

[ \frac{(1+\sin\To verify the identity ( \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} = 2 \sec^2(\theta) ), we can start with the left-hand side of the equation and simplify it step by step.

[ \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} ]

To combine these fractions, we find a common denominator:

[ = \frac{(1 + \sin(\theta)) + (1 - \sin(\theta))}{(1 - \sin^To verify the identity ( \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} = 2\sec^2\theta ), we can start with the left-hand side (LHS) of the equation and manipulate it until it matches the right-hand side (RHS).

Starting with LHS: [ \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} ]

To combine these fractions, we need a common denominator. The common denominator here is ((1-\sin\theta)(1+\sin\theta)).

[ \frac{(1+\sin\thetaTo verify the identity ( \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} = 2 \sec^2(\theta) ), we can start with the left-hand side of the equation and simplify it step by step.

[ \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} ]

To combine these fractions, we find a common denominator:

[ = \frac{(1 + \sin(\theta)) + (1 - \sin(\theta))}{(1 - \sin^2To verify the identity ( \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} = 2\sec^2\theta ), we can start with the left-hand side (LHS) of the equation and manipulate it until it matches the right-hand side (RHS).

Starting with LHS: [ \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} ]

To combine these fractions, we need a common denominator. The common denominator here is ((1-\sin\theta)(1+\sin\theta)).

[ \frac{(1+\sin\theta)To verify the identity ( \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} = 2 \sec^2(\theta) ), we can start with the left-hand side of the equation and simplify it step by step.

[ \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} ]

To combine these fractions, we find a common denominator:

[ = \frac{(1 + \sin(\theta)) + (1 - \sin(\theta))}{(1 - \sin^2(\To verify the identity ( \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} = 2\sec^2\theta ), we can start with the left-hand side (LHS) of the equation and manipulate it until it matches the right-hand side (RHS).

Starting with LHS: [ \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} ]

To combine these fractions, we need a common denominator. The common denominator here is ((1-\sin\theta)(1+\sin\theta)).

[ \frac{(1+\sin\theta) +To verify the identity ( \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} = 2 \sec^2(\theta) ), we can start with the left-hand side of the equation and simplify it step by step.

[ \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} ]

To combine these fractions, we find a common denominator:

[ = \frac{(1 + \sin(\theta)) + (1 - \sin(\theta))}{(1 - \sin^2(\thetaTo verify the identity ( \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} = 2\sec^2\theta ), we can start with the left-hand side (LHS) of the equation and manipulate it until it matches the right-hand side (RHS).

Starting with LHS: [ \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} ]

To combine these fractions, we need a common denominator. The common denominator here is ((1-\sin\theta)(1+\sin\theta)).

[ \frac{(1+\sin\theta) + (1To verify the identity ( \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} = 2 \sec^2(\theta) ), we can start with the left-hand side of the equation and simplify it step by step.

[ \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} ]

To combine these fractions, we find a common denominator:

[ = \frac{(1 + \sin(\theta)) + (1 - \sin(\theta))}{(1 - \sin^2(\theta))To verify the identity ( \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} = 2\sec^2\theta ), we can start with the left-hand side (LHS) of the equation and manipulate it until it matches the right-hand side (RHS).

Starting with LHS: [ \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} ]

To combine these fractions, we need a common denominator. The common denominator here is ((1-\sin\theta)(1+\sin\theta)).

[ \frac{(1+\sin\theta) + (1-\To verify the identity ( \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} = 2 \sec^2(\theta) ), we can start with the left-hand side of the equation and simplify it step by step.

[ \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} ]

To combine these fractions, we find a common denominator:

[ = \frac{(1 + \sin(\theta)) + (1 - \sin(\theta))}{(1 - \sin^2(\theta))}To verify the identity ( \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} = 2\sec^2\theta ), we can start with the left-hand side (LHS) of the equation and manipulate it until it matches the right-hand side (RHS).

Starting with LHS: [ \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} ]

To combine these fractions, we need a common denominator. The common denominator here is ((1-\sin\theta)(1+\sin\theta)).

[ \frac{(1+\sin\theta) + (1-\sin\To verify the identity ( \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} = 2 \sec^2(\theta) ), we can start with the left-hand side of the equation and simplify it step by step.

[ \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} ]

To combine these fractions, we find a common denominator:

[ = \frac{(1 + \sin(\theta)) + (1 - \sin(\theta))}{(1 - \sin^2(\theta))} \To verify the identity ( \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} = 2\sec^2\theta ), we can start with the left-hand side (LHS) of the equation and manipulate it until it matches the right-hand side (RHS).

Starting with LHS: [ \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} ]

To combine these fractions, we need a common denominator. The common denominator here is ((1-\sin\theta)(1+\sin\theta)).

[ \frac{(1+\sin\theta) + (1-\sin\theta)}{(To verify the identity ( \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} = 2 \sec^2(\theta) ), we can start with the left-hand side of the equation and simplify it step by step.

[ \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} ]

To combine these fractions, we find a common denominator:

[ = \frac{(1 + \sin(\theta)) + (1 - \sin(\theta))}{(1 - \sin^2(\theta))} ]

To verify the identity ( \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} = 2\sec^2\theta ), we can start with the left-hand side (LHS) of the equation and manipulate it until it matches the right-hand side (RHS).

Starting with LHS: [ \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} ]

To combine these fractions, we need a common denominator. The common denominator here is ((1-\sin\theta)(1+\sin\theta)).

[ \frac{(1+\sin\theta) + (1-\sin\theta)}{(1To verify the identity ( \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} = 2 \sec^2(\theta) ), we can start with the left-hand side of the equation and simplify it step by step.

[ \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} ]

To combine these fractions, we find a common denominator:

[ = \frac{(1 + \sin(\theta)) + (1 - \sin(\theta))}{(1 - \sin^2(\theta))} ]

NowTo verify the identity ( \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} = 2\sec^2\theta ), we can start with the left-hand side (LHS) of the equation and manipulate it until it matches the right-hand side (RHS).

Starting with LHS: [ \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} ]

To combine these fractions, we need a common denominator. The common denominator here is ((1-\sin\theta)(1+\sin\theta)).

[ \frac{(1+\sin\theta) + (1-\sin\theta)}{(1-\sin\To verify the identity ( \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} = 2 \sec^2(\theta) ), we can start with the left-hand side of the equation and simplify it step by step.

[ \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} ]

To combine these fractions, we find a common denominator:

[ = \frac{(1 + \sin(\theta)) + (1 - \sin(\theta))}{(1 - \sin^2(\theta))} ]

Now,To verify the identity ( \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} = 2\sec^2\theta ), we can start with the left-hand side (LHS) of the equation and manipulate it until it matches the right-hand side (RHS).

Starting with LHS: [ \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} ]

To combine these fractions, we need a common denominator. The common denominator here is ((1-\sin\theta)(1+\sin\theta)).

[ \frac{(1+\sin\theta) + (1-\sin\theta)}{(1-\sin\thetaTo verify the identity ( \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} = 2 \sec^2(\theta) ), we can start with the left-hand side of the equation and simplify it step by step.

[ \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} ]

To combine these fractions, we find a common denominator:

[ = \frac{(1 + \sin(\theta)) + (1 - \sin(\theta))}{(1 - \sin^2(\theta))} ]

Now, sinceTo verify the identity ( \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} = 2\sec^2\theta ), we can start with the left-hand side (LHS) of the equation and manipulate it until it matches the right-hand side (RHS).

Starting with LHS: [ \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} ]

To combine these fractions, we need a common denominator. The common denominator here is ((1-\sin\theta)(1+\sin\theta)).

[ \frac{(1+\sin\theta) + (1-\sin\theta)}{(1-\sin\theta)(To verify the identity ( \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} = 2 \sec^2(\theta) ), we can start with the left-hand side of the equation and simplify it step by step.

[ \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} ]

To combine these fractions, we find a common denominator:

[ = \frac{(1 + \sin(\theta)) + (1 - \sin(\theta))}{(1 - \sin^2(\theta))} ]

Now, since (To verify the identity ( \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} = 2\sec^2\theta ), we can start with the left-hand side (LHS) of the equation and manipulate it until it matches the right-hand side (RHS).

Starting with LHS: [ \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} ]

To combine these fractions, we need a common denominator. The common denominator here is ((1-\sin\theta)(1+\sin\theta)).

[ \frac{(1+\sin\theta) + (1-\sin\theta)}{(1-\sin\theta)(1To verify the identity ( \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} = 2 \sec^2(\theta) ), we can start with the left-hand side of the equation and simplify it step by step.

[ \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} ]

To combine these fractions, we find a common denominator:

[ = \frac{(1 + \sin(\theta)) + (1 - \sin(\theta))}{(1 - \sin^2(\theta))} ]

Now, since ( To verify the identity ( \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} = 2\sec^2\theta ), we can start with the left-hand side (LHS) of the equation and manipulate it until it matches the right-hand side (RHS).

Starting with LHS: [ \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} ]

To combine these fractions, we need a common denominator. The common denominator here is ((1-\sin\theta)(1+\sin\theta)).

[ \frac{(1+\sin\theta) + (1-\sin\theta)}{(1-\sin\theta)(1+\To verify the identity ( \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} = 2 \sec^2(\theta) ), we can start with the left-hand side of the equation and simplify it step by step.

[ \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} ]

To combine these fractions, we find a common denominator:

[ = \frac{(1 + \sin(\theta)) + (1 - \sin(\theta))}{(1 - \sin^2(\theta))} ]

Now, since ( 1To verify the identity ( \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} = 2\sec^2\theta ), we can start with the left-hand side (LHS) of the equation and manipulate it until it matches the right-hand side (RHS).

Starting with LHS: [ \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} ]

To combine these fractions, we need a common denominator. The common denominator here is ((1-\sin\theta)(1+\sin\theta)).

[ \frac{(1+\sin\theta) + (1-\sin\theta)}{(1-\sin\theta)(1+\sinTo verify the identity ( \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} = 2 \sec^2(\theta) ), we can start with the left-hand side of the equation and simplify it step by step.

[ \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} ]

To combine these fractions, we find a common denominator:

[ = \frac{(1 + \sin(\theta)) + (1 - \sin(\theta))}{(1 - \sin^2(\theta))} ]

Now, since ( 1 -To verify the identity ( \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} = 2\sec^2\theta ), we can start with the left-hand side (LHS) of the equation and manipulate it until it matches the right-hand side (RHS).

Starting with LHS: [ \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} ]

To combine these fractions, we need a common denominator. The common denominator here is ((1-\sin\theta)(1+\sin\theta)).

[ \frac{(1+\sin\theta) + (1-\sin\theta)}{(1-\sin\theta)(1+\sin\To verify the identity ( \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} = 2 \sec^2(\theta) ), we can start with the left-hand side of the equation and simplify it step by step.

[ \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} ]

To combine these fractions, we find a common denominator:

[ = \frac{(1 + \sin(\theta)) + (1 - \sin(\theta))}{(1 - \sin^2(\theta))} ]

Now, since ( 1 - \To verify the identity ( \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} = 2\sec^2\theta ), we can start with the left-hand side (LHS) of the equation and manipulate it until it matches the right-hand side (RHS).

Starting with LHS: [ \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} ]

To combine these fractions, we need a common denominator. The common denominator here is ((1-\sin\theta)(1+\sin\theta)).

[ \frac{(1+\sin\theta) + (1-\sin\theta)}{(1-\sin\theta)(1+\sin\thetaTo verify the identity ( \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} = 2 \sec^2(\theta) ), we can start with the left-hand side of the equation and simplify it step by step.

[ \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} ]

To combine these fractions, we find a common denominator:

[ = \frac{(1 + \sin(\theta)) + (1 - \sin(\theta))}{(1 - \sin^2(\theta))} ]

Now, since ( 1 - \sinTo verify the identity ( \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} = 2\sec^2\theta ), we can start with the left-hand side (LHS) of the equation and manipulate it until it matches the right-hand side (RHS).

Starting with LHS: [ \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} ]

To combine these fractions, we need a common denominator. The common denominator here is ((1-\sin\theta)(1+\sin\theta)).

[ \frac{(1+\sin\theta) + (1-\sin\theta)}{(1-\sin\theta)(1+\sin\theta)}To verify the identity ( \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} = 2 \sec^2(\theta) ), we can start with the left-hand side of the equation and simplify it step by step.

[ \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} ]

To combine these fractions, we find a common denominator:

[ = \frac{(1 + \sin(\theta)) + (1 - \sin(\theta))}{(1 - \sin^2(\theta))} ]

Now, since ( 1 - \sin^To verify the identity ( \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} = 2\sec^2\theta ), we can start with the left-hand side (LHS) of the equation and manipulate it until it matches the right-hand side (RHS).

Starting with LHS: [ \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} ]

To combine these fractions, we need a common denominator. The common denominator here is ((1-\sin\theta)(1+\sin\theta)).

[ \frac{(1+\sin\theta) + (1-\sin\theta)}{(1-\sin\theta)(1+\sin\theta)} \To verify the identity ( \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} = 2 \sec^2(\theta) ), we can start with the left-hand side of the equation and simplify it step by step.

[ \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} ]

To combine these fractions, we find a common denominator:

[ = \frac{(1 + \sin(\theta)) + (1 - \sin(\theta))}{(1 - \sin^2(\theta))} ]

Now, since ( 1 - \sin^2(\theta) = \cos^2(\theta) \To verify the identity ( \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} = 2\sec^2\theta ), we can start with the left-hand side (LHS) of the equation and manipulate it until it matches the right-hand side (RHS).

Starting with LHS: [ \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} ]

To combine these fractions, we need a common denominator. The common denominator here is ((1-\sin\theta)(1+\sin\theta)).

[ \frac{(1+\sin\theta) + (1-\sin\theta)}{(1-\sin\theta)(1+\sin\theta)} ]

To verify the identity ( \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} = 2 \sec^2(\theta) ), we can start with the left-hand side of the equation and simplify it step by step.

[ \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} ]

To combine these fractions, we find a common denominator:

[ = \frac{(1 + \sin(\theta)) + (1 - \sin(\theta))}{(1 - \sin^2(\theta))} ]

Now, since ( 1 - \sin^2(\theta) = \cos^2(\theta) ),To verify the identity ( \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} = 2\sec^2\theta ), we can start with the left-hand side (LHS) of the equation and manipulate it until it matches the right-hand side (RHS).

Starting with LHS: [ \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} ]

To combine these fractions, we need a common denominator. The common denominator here is ((1-\sin\theta)(1+\sin\theta)).

[ \frac{(1+\sin\theta) + (1-\sin\theta)}{(1-\sin\theta)(1+\sin\theta)} ]

STo verify the identity ( \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} = 2 \sec^2(\theta) ), we can start with the left-hand side of the equation and simplify it step by step.

[ \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} ]

To combine these fractions, we find a common denominator:

[ = \frac{(1 + \sin(\theta)) + (1 - \sin(\theta))}{(1 - \sin^2(\theta))} ]

Now, since ( 1 - \sin^2(\theta) = \cos^2(\theta) ), weTo verify the identity ( \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} = 2\sec^2\theta ), we can start with the left-hand side (LHS) of the equation and manipulate it until it matches the right-hand side (RHS).

Starting with LHS: [ \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} ]

To combine these fractions, we need a common denominator. The common denominator here is ((1-\sin\theta)(1+\sin\theta)).

[ \frac{(1+\sin\theta) + (1-\sin\theta)}{(1-\sin\theta)(1+\sin\theta)} ]

SimplifyTo verify the identity ( \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} = 2 \sec^2(\theta) ), we can start with the left-hand side of the equation and simplify it step by step.

[ \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} ]

To combine these fractions, we find a common denominator:

[ = \frac{(1 + \sin(\theta)) + (1 - \sin(\theta))}{(1 - \sin^2(\theta))} ]

Now, since ( 1 - \sin^2(\theta) = \cos^2(\theta) ), we canTo verify the identity ( \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} = 2\sec^2\theta ), we can start with the left-hand side (LHS) of the equation and manipulate it until it matches the right-hand side (RHS).

Starting with LHS: [ \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} ]

To combine these fractions, we need a common denominator. The common denominator here is ((1-\sin\theta)(1+\sin\theta)).

[ \frac{(1+\sin\theta) + (1-\sin\theta)}{(1-\sin\theta)(1+\sin\theta)} ]

Simplify theTo verify the identity ( \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} = 2 \sec^2(\theta) ), we can start with the left-hand side of the equation and simplify it step by step.

[ \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} ]

To combine these fractions, we find a common denominator:

[ = \frac{(1 + \sin(\theta)) + (1 - \sin(\theta))}{(1 - \sin^2(\theta))} ]

Now, since ( 1 - \sin^2(\theta) = \cos^2(\theta) ), we can simplifyTo verify the identity ( \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} = 2\sec^2\theta ), we can start with the left-hand side (LHS) of the equation and manipulate it until it matches the right-hand side (RHS).

Starting with LHS: [ \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} ]

To combine these fractions, we need a common denominator. The common denominator here is ((1-\sin\theta)(1+\sin\theta)).

[ \frac{(1+\sin\theta) + (1-\sin\theta)}{(1-\sin\theta)(1+\sin\theta)} ]

Simplify the numeratorTo verify the identity ( \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} = 2 \sec^2(\theta) ), we can start with the left-hand side of the equation and simplify it step by step.

[ \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} ]

To combine these fractions, we find a common denominator:

[ = \frac{(1 + \sin(\theta)) + (1 - \sin(\theta))}{(1 - \sin^2(\theta))} ]

Now, since ( 1 - \sin^2(\theta) = \cos^2(\theta) ), we can simplify furtherTo verify the identity ( \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} = 2\sec^2\theta ), we can start with the left-hand side (LHS) of the equation and manipulate it until it matches the right-hand side (RHS).

Starting with LHS: [ \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} ]

To combine these fractions, we need a common denominator. The common denominator here is ((1-\sin\theta)(1+\sin\theta)).

[ \frac{(1+\sin\theta) + (1-\sin\theta)}{(1-\sin\theta)(1+\sin\theta)} ]

Simplify the numerator:

To verify the identity ( \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} = 2 \sec^2(\theta) ), we can start with the left-hand side of the equation and simplify it step by step.

[ \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} ]

To combine these fractions, we find a common denominator:

[ = \frac{(1 + \sin(\theta)) + (1 - \sin(\theta))}{(1 - \sin^2(\theta))} ]

Now, since ( 1 - \sin^2(\theta) = \cos^2(\theta) ), we can simplify further:

To verify the identity ( \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} = 2\sec^2\theta ), we can start with the left-hand side (LHS) of the equation and manipulate it until it matches the right-hand side (RHS).

Starting with LHS: [ \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} ]

To combine these fractions, we need a common denominator. The common denominator here is ((1-\sin\theta)(1+\sin\theta)).

[ \frac{(1+\sin\theta) + (1-\sin\theta)}{(1-\sin\theta)(1+\sin\theta)} ]

Simplify the numerator:

[To verify the identity ( \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} = 2 \sec^2(\theta) ), we can start with the left-hand side of the equation and simplify it step by step.

[ \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} ]

To combine these fractions, we find a common denominator:

[ = \frac{(1 + \sin(\theta)) + (1 - \sin(\theta))}{(1 - \sin^2(\theta))} ]

Now, since ( 1 - \sin^2(\theta) = \cos^2(\theta) ), we can simplify further:

[To verify the identity ( \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} = 2\sec^2\theta ), we can start with the left-hand side (LHS) of the equation and manipulate it until it matches the right-hand side (RHS).

Starting with LHS: [ \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} ]

To combine these fractions, we need a common denominator. The common denominator here is ((1-\sin\theta)(1+\sin\theta)).

[ \frac{(1+\sin\theta) + (1-\sin\theta)}{(1-\sin\theta)(1+\sin\theta)} ]

Simplify the numerator:

[ \To verify the identity ( \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} = 2 \sec^2(\theta) ), we can start with the left-hand side of the equation and simplify it step by step.

[ \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} ]

To combine these fractions, we find a common denominator:

[ = \frac{(1 + \sin(\theta)) + (1 - \sin(\theta))}{(1 - \sin^2(\theta))} ]

Now, since ( 1 - \sin^2(\theta) = \cos^2(\theta) ), we can simplify further:

[ =To verify the identity ( \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} = 2\sec^2\theta ), we can start with the left-hand side (LHS) of the equation and manipulate it until it matches the right-hand side (RHS).

Starting with LHS: [ \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} ]

To combine these fractions, we need a common denominator. The common denominator here is ((1-\sin\theta)(1+\sin\theta)).

[ \frac{(1+\sin\theta) + (1-\sin\theta)}{(1-\sin\theta)(1+\sin\theta)} ]

Simplify the numerator:

[ \fracTo verify the identity ( \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} = 2 \sec^2(\theta) ), we can start with the left-hand side of the equation and simplify it step by step.

[ \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} ]

To combine these fractions, we find a common denominator:

[ = \frac{(1 + \sin(\theta)) + (1 - \sin(\theta))}{(1 - \sin^2(\theta))} ]

Now, since ( 1 - \sin^2(\theta) = \cos^2(\theta) ), we can simplify further:

[ = \To verify the identity ( \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} = 2\sec^2\theta ), we can start with the left-hand side (LHS) of the equation and manipulate it until it matches the right-hand side (RHS).

Starting with LHS: [ \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} ]

To combine these fractions, we need a common denominator. The common denominator here is ((1-\sin\theta)(1+\sin\theta)).

[ \frac{(1+\sin\theta) + (1-\sin\theta)}{(1-\sin\theta)(1+\sin\theta)} ]

Simplify the numerator:

[ \frac{1To verify the identity ( \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} = 2 \sec^2(\theta) ), we can start with the left-hand side of the equation and simplify it step by step.

[ \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} ]

To combine these fractions, we find a common denominator:

[ = \frac{(1 + \sin(\theta)) + (1 - \sin(\theta))}{(1 - \sin^2(\theta))} ]

Now, since ( 1 - \sin^2(\theta) = \cos^2(\theta) ), we can simplify further:

[ = \frac{1To verify the identity ( \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} = 2\sec^2\theta ), we can start with the left-hand side (LHS) of the equation and manipulate it until it matches the right-hand side (RHS).

Starting with LHS: [ \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} ]

To combine these fractions, we need a common denominator. The common denominator here is ((1-\sin\theta)(1+\sin\theta)).

[ \frac{(1+\sin\theta) + (1-\sin\theta)}{(1-\sin\theta)(1+\sin\theta)} ]

Simplify the numerator:

[ \frac{1+\To verify the identity ( \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} = 2 \sec^2(\theta) ), we can start with the left-hand side of the equation and simplify it step by step.

[ \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} ]

To combine these fractions, we find a common denominator:

[ = \frac{(1 + \sin(\theta)) + (1 - \sin(\theta))}{(1 - \sin^2(\theta))} ]

Now, since ( 1 - \sin^2(\theta) = \cos^2(\theta) ), we can simplify further:

[ = \frac{1 +To verify the identity ( \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} = 2\sec^2\theta ), we can start with the left-hand side (LHS) of the equation and manipulate it until it matches the right-hand side (RHS).

Starting with LHS: [ \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} ]

To combine these fractions, we need a common denominator. The common denominator here is ((1-\sin\theta)(1+\sin\theta)).

[ \frac{(1+\sin\theta) + (1-\sin\theta)}{(1-\sin\theta)(1+\sin\theta)} ]

Simplify the numerator:

[ \frac{1+\sinTo verify the identity ( \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} = 2 \sec^2(\theta) ), we can start with the left-hand side of the equation and simplify it step by step.

[ \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} ]

To combine these fractions, we find a common denominator:

[ = \frac{(1 + \sin(\theta)) + (1 - \sin(\theta))}{(1 - \sin^2(\theta))} ]

Now, since ( 1 - \sin^2(\theta) = \cos^2(\theta) ), we can simplify further:

[ = \frac{1 + \To verify the identity ( \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} = 2\sec^2\theta ), we can start with the left-hand side (LHS) of the equation and manipulate it until it matches the right-hand side (RHS).

Starting with LHS: [ \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} ]

To combine these fractions, we need a common denominator. The common denominator here is ((1-\sin\theta)(1+\sin\theta)).

[ \frac{(1+\sin\theta) + (1-\sin\theta)}{(1-\sin\theta)(1+\sin\theta)} ]

Simplify the numerator:

[ \frac{1+\sin\thetaTo verify the identity ( \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} = 2 \sec^2(\theta) ), we can start with the left-hand side of the equation and simplify it step by step.

[ \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} ]

To combine these fractions, we find a common denominator:

[ = \frac{(1 + \sin(\theta)) + (1 - \sin(\theta))}{(1 - \sin^2(\theta))} ]

Now, since ( 1 - \sin^2(\theta) = \cos^2(\theta) ), we can simplify further:

[ = \frac{1 + \sinTo verify the identity ( \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} = 2\sec^2\theta ), we can start with the left-hand side (LHS) of the equation and manipulate it until it matches the right-hand side (RHS).

Starting with LHS: [ \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} ]

To combine these fractions, we need a common denominator. The common denominator here is ((1-\sin\theta)(1+\sin\theta)).

[ \frac{(1+\sin\theta) + (1-\sin\theta)}{(1-\sin\theta)(1+\sin\theta)} ]

Simplify the numerator:

[ \frac{1+\sin\theta +To verify the identity ( \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} = 2 \sec^2(\theta) ), we can start with the left-hand side of the equation and simplify it step by step.

[ \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} ]

To combine these fractions, we find a common denominator:

[ = \frac{(1 + \sin(\theta)) + (1 - \sin(\theta))}{(1 - \sin^2(\theta))} ]

Now, since ( 1 - \sin^2(\theta) = \cos^2(\theta) ), we can simplify further:

[ = \frac{1 + \sin(\To verify the identity ( \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} = 2\sec^2\theta ), we can start with the left-hand side (LHS) of the equation and manipulate it until it matches the right-hand side (RHS).

Starting with LHS: [ \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} ]

To combine these fractions, we need a common denominator. The common denominator here is ((1-\sin\theta)(1+\sin\theta)).

[ \frac{(1+\sin\theta) + (1-\sin\theta)}{(1-\sin\theta)(1+\sin\theta)} ]

Simplify the numerator:

[ \frac{1+\sin\theta + To verify the identity ( \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} = 2 \sec^2(\theta) ), we can start with the left-hand side of the equation and simplify it step by step.

[ \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} ]

To combine these fractions, we find a common denominator:

[ = \frac{(1 + \sin(\theta)) + (1 - \sin(\theta))}{(1 - \sin^2(\theta))} ]

Now, since ( 1 - \sin^2(\theta) = \cos^2(\theta) ), we can simplify further:

[ = \frac{1 + \sin(\thetaTo verify the identity ( \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} = 2\sec^2\theta ), we can start with the left-hand side (LHS) of the equation and manipulate it until it matches the right-hand side (RHS).

Starting with LHS: [ \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} ]

To combine these fractions, we need a common denominator. The common denominator here is ((1-\sin\theta)(1+\sin\theta)).

[ \frac{(1+\sin\theta) + (1-\sin\theta)}{(1-\sin\theta)(1+\sin\theta)} ]

Simplify the numerator:

[ \frac{1+\sin\theta + 1To verify the identity ( \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} = 2 \sec^2(\theta) ), we can start with the left-hand side of the equation and simplify it step by step.

[ \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} ]

To combine these fractions, we find a common denominator:

[ = \frac{(1 + \sin(\theta)) + (1 - \sin(\theta))}{(1 - \sin^2(\theta))} ]

Now, since ( 1 - \sin^2(\theta) = \cos^2(\theta) ), we can simplify further:

[ = \frac{1 + \sin(\theta)To verify the identity ( \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} = 2\sec^2\theta ), we can start with the left-hand side (LHS) of the equation and manipulate it until it matches the right-hand side (RHS).

Starting with LHS: [ \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} ]

To combine these fractions, we need a common denominator. The common denominator here is ((1-\sin\theta)(1+\sin\theta)).

[ \frac{(1+\sin\theta) + (1-\sin\theta)}{(1-\sin\theta)(1+\sin\theta)} ]

Simplify the numerator:

[ \frac{1+\sin\theta + 1-\To verify the identity ( \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} = 2 \sec^2(\theta) ), we can start with the left-hand side of the equation and simplify it step by step.

[ \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} ]

To combine these fractions, we find a common denominator:

[ = \frac{(1 + \sin(\theta)) + (1 - \sin(\theta))}{(1 - \sin^2(\theta))} ]

Now, since ( 1 - \sin^2(\theta) = \cos^2(\theta) ), we can simplify further:

[ = \frac{1 + \sin(\theta) + To verify the identity ( \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} = 2\sec^2\theta ), we can start with the left-hand side (LHS) of the equation and manipulate it until it matches the right-hand side (RHS).

Starting with LHS: [ \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} ]

To combine these fractions, we need a common denominator. The common denominator here is ((1-\sin\theta)(1+\sin\theta)).

[ \frac{(1+\sin\theta) + (1-\sin\theta)}{(1-\sin\theta)(1+\sin\theta)} ]

Simplify the numerator:

[ \frac{1+\sin\theta + 1-\sin\To verify the identity ( \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} = 2 \sec^2(\theta) ), we can start with the left-hand side of the equation and simplify it step by step.

[ \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} ]

To combine these fractions, we find a common denominator:

[ = \frac{(1 + \sin(\theta)) + (1 - \sin(\theta))}{(1 - \sin^2(\theta))} ]

Now, since ( 1 - \sin^2(\theta) = \cos^2(\theta) ), we can simplify further:

[ = \frac{1 + \sin(\theta) + 1To verify the identity ( \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} = 2\sec^2\theta ), we can start with the left-hand side (LHS) of the equation and manipulate it until it matches the right-hand side (RHS).

Starting with LHS: [ \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} ]

To combine these fractions, we need a common denominator. The common denominator here is ((1-\sin\theta)(1+\sin\theta)).

[ \frac{(1+\sin\theta) + (1-\sin\theta)}{(1-\sin\theta)(1+\sin\theta)} ]

Simplify the numerator:

[ \frac{1+\sin\theta + 1-\sin\thetaTo verify the identity ( \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} = 2 \sec^2(\theta) ), we can start with the left-hand side of the equation and simplify it step by step.

[ \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} ]

To combine these fractions, we find a common denominator:

[ = \frac{(1 + \sin(\theta)) + (1 - \sin(\theta))}{(1 - \sin^2(\theta))} ]

Now, since ( 1 - \sin^2(\theta) = \cos^2(\theta) ), we can simplify further:

[ = \frac{1 + \sin(\theta) + 1 -To verify the identity ( \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} = 2\sec^2\theta ), we can start with the left-hand side (LHS) of the equation and manipulate it until it matches the right-hand side (RHS).

Starting with LHS: [ \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} ]

To combine these fractions, we need a common denominator. The common denominator here is ((1-\sin\theta)(1+\sin\theta)).

[ \frac{(1+\sin\theta) + (1-\sin\theta)}{(1-\sin\theta)(1+\sin\theta)} ]

Simplify the numerator:

[ \frac{1+\sin\theta + 1-\sin\theta}{To verify the identity ( \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} = 2 \sec^2(\theta) ), we can start with the left-hand side of the equation and simplify it step by step.

[ \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} ]

To combine these fractions, we find a common denominator:

[ = \frac{(1 + \sin(\theta)) + (1 - \sin(\theta))}{(1 - \sin^2(\theta))} ]

Now, since ( 1 - \sin^2(\theta) = \cos^2(\theta) ), we can simplify further:

[ = \frac{1 + \sin(\theta) + 1 - \To verify the identity ( \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} = 2\sec^2\theta ), we can start with the left-hand side (LHS) of the equation and manipulate it until it matches the right-hand side (RHS).

Starting with LHS: [ \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} ]

To combine these fractions, we need a common denominator. The common denominator here is ((1-\sin\theta)(1+\sin\theta)).

[ \frac{(1+\sin\theta) + (1-\sin\theta)}{(1-\sin\theta)(1+\sin\theta)} ]

Simplify the numerator:

[ \frac{1+\sin\theta + 1-\sin\theta}{(To verify the identity ( \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} = 2 \sec^2(\theta) ), we can start with the left-hand side of the equation and simplify it step by step.

[ \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} ]

To combine these fractions, we find a common denominator:

[ = \frac{(1 + \sin(\theta)) + (1 - \sin(\theta))}{(1 - \sin^2(\theta))} ]

Now, since ( 1 - \sin^2(\theta) = \cos^2(\theta) ), we can simplify further:

[ = \frac{1 + \sin(\theta) + 1 - \sinTo verify the identity ( \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} = 2\sec^2\theta ), we can start with the left-hand side (LHS) of the equation and manipulate it until it matches the right-hand side (RHS).

Starting with LHS: [ \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} ]

To combine these fractions, we need a common denominator. The common denominator here is ((1-\sin\theta)(1+\sin\theta)).

[ \frac{(1+\sin\theta) + (1-\sin\theta)}{(1-\sin\theta)(1+\sin\theta)} ]

Simplify the numerator:

[ \frac{1+\sin\theta + 1-\sin\theta}{(1To verify the identity ( \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} = 2 \sec^2(\theta) ), we can start with the left-hand side of the equation and simplify it step by step.

[ \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} ]

To combine these fractions, we find a common denominator:

[ = \frac{(1 + \sin(\theta)) + (1 - \sin(\theta))}{(1 - \sin^2(\theta))} ]

Now, since ( 1 - \sin^2(\theta) = \cos^2(\theta) ), we can simplify further:

[ = \frac{1 + \sin(\theta) + 1 - \sin(\To verify the identity ( \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} = 2\sec^2\theta ), we can start with the left-hand side (LHS) of the equation and manipulate it until it matches the right-hand side (RHS).

Starting with LHS: [ \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} ]

To combine these fractions, we need a common denominator. The common denominator here is ((1-\sin\theta)(1+\sin\theta)).

[ \frac{(1+\sin\theta) + (1-\sin\theta)}{(1-\sin\theta)(1+\sin\theta)} ]

Simplify the numerator:

[ \frac{1+\sin\theta + 1-\sin\theta}{(1-\To verify the identity ( \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} = 2 \sec^2(\theta) ), we can start with the left-hand side of the equation and simplify it step by step.

[ \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} ]

To combine these fractions, we find a common denominator:

[ = \frac{(1 + \sin(\theta)) + (1 - \sin(\theta))}{(1 - \sin^2(\theta))} ]

Now, since ( 1 - \sin^2(\theta) = \cos^2(\theta) ), we can simplify further:

[ = \frac{1 + \sin(\theta) + 1 - \sin(\thetaTo verify the identity ( \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} = 2\sec^2\theta ), we can start with the left-hand side (LHS) of the equation and manipulate it until it matches the right-hand side (RHS).

Starting with LHS: [ \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} ]

To combine these fractions, we need a common denominator. The common denominator here is ((1-\sin\theta)(1+\sin\theta)).

[ \frac{(1+\sin\theta) + (1-\sin\theta)}{(1-\sin\theta)(1+\sin\theta)} ]

Simplify the numerator:

[ \frac{1+\sin\theta + 1-\sin\theta}{(1-\sin\To verify the identity ( \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} = 2 \sec^2(\theta) ), we can start with the left-hand side of the equation and simplify it step by step.

[ \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} ]

To combine these fractions, we find a common denominator:

[ = \frac{(1 + \sin(\theta)) + (1 - \sin(\theta))}{(1 - \sin^2(\theta))} ]

Now, since ( 1 - \sin^2(\theta) = \cos^2(\theta) ), we can simplify further:

[ = \frac{1 + \sin(\theta) + 1 - \sin(\theta)}To verify the identity ( \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} = 2\sec^2\theta ), we can start with the left-hand side (LHS) of the equation and manipulate it until it matches the right-hand side (RHS).

Starting with LHS: [ \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} ]

To combine these fractions, we need a common denominator. The common denominator here is ((1-\sin\theta)(1+\sin\theta)).

[ \frac{(1+\sin\theta) + (1-\sin\theta)}{(1-\sin\theta)(1+\sin\theta)} ]

Simplify the numerator:

[ \frac{1+\sin\theta + 1-\sin\theta}{(1-\sin\thetaTo verify the identity ( \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} = 2 \sec^2(\theta) ), we can start with the left-hand side of the equation and simplify it step by step.

[ \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} ]

To combine these fractions, we find a common denominator:

[ = \frac{(1 + \sin(\theta)) + (1 - \sin(\theta))}{(1 - \sin^2(\theta))} ]

Now, since ( 1 - \sin^2(\theta) = \cos^2(\theta) ), we can simplify further:

[ = \frac{1 + \sin(\theta) + 1 - \sin(\theta)}{\cosTo verify the identity ( \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} = 2\sec^2\theta ), we can start with the left-hand side (LHS) of the equation and manipulate it until it matches the right-hand side (RHS).

Starting with LHS: [ \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} ]

To combine these fractions, we need a common denominator. The common denominator here is ((1-\sin\theta)(1+\sin\theta)).

[ \frac{(1+\sin\theta) + (1-\sin\theta)}{(1-\sin\theta)(1+\sin\theta)} ]

Simplify the numerator:

[ \frac{1+\sin\theta + 1-\sin\theta}{(1-\sin\theta)(1To verify the identity ( \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} = 2 \sec^2(\theta) ), we can start with the left-hand side of the equation and simplify it step by step.

[ \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} ]

To combine these fractions, we find a common denominator:

[ = \frac{(1 + \sin(\theta)) + (1 - \sin(\theta))}{(1 - \sin^2(\theta))} ]

Now, since ( 1 - \sin^2(\theta) = \cos^2(\theta) ), we can simplify further:

[ = \frac{1 + \sin(\theta) + 1 - \sin(\theta)}{\cos^2To verify the identity ( \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} = 2\sec^2\theta ), we can start with the left-hand side (LHS) of the equation and manipulate it until it matches the right-hand side (RHS).

Starting with LHS: [ \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} ]

To combine these fractions, we need a common denominator. The common denominator here is ((1-\sin\theta)(1+\sin\theta)).

[ \frac{(1+\sin\theta) + (1-\sin\theta)}{(1-\sin\theta)(1+\sin\theta)} ]

Simplify the numerator:

[ \frac{1+\sin\theta + 1-\sin\theta}{(1-\sin\theta)(1+\To verify the identity ( \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} = 2 \sec^2(\theta) ), we can start with the left-hand side of the equation and simplify it step by step.

[ \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} ]

To combine these fractions, we find a common denominator:

[ = \frac{(1 + \sin(\theta)) + (1 - \sin(\theta))}{(1 - \sin^2(\theta))} ]

Now, since ( 1 - \sin^2(\theta) = \cos^2(\theta) ), we can simplify further:

[ = \frac{1 + \sin(\theta) + 1 - \sin(\theta)}{\cos^2(\To verify the identity ( \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} = 2\sec^2\theta ), we can start with the left-hand side (LHS) of the equation and manipulate it until it matches the right-hand side (RHS).

Starting with LHS: [ \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} ]

To combine these fractions, we need a common denominator. The common denominator here is ((1-\sin\theta)(1+\sin\theta)).

[ \frac{(1+\sin\theta) + (1-\sin\theta)}{(1-\sin\theta)(1+\sin\theta)} ]

Simplify the numerator:

[ \frac{1+\sin\theta + 1-\sin\theta}{(1-\sin\theta)(1+\sinTo verify the identity ( \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} = 2 \sec^2(\theta) ), we can start with the left-hand side of the equation and simplify it step by step.

[ \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} ]

To combine these fractions, we find a common denominator:

[ = \frac{(1 + \sin(\theta)) + (1 - \sin(\theta))}{(1 - \sin^2(\theta))} ]

Now, since ( 1 - \sin^2(\theta) = \cos^2(\theta) ), we can simplify further:

[ = \frac{1 + \sin(\theta) + 1 - \sin(\theta)}{\cos^2(\thetaTo verify the identity ( \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} = 2\sec^2\theta ), we can start with the left-hand side (LHS) of the equation and manipulate it until it matches the right-hand side (RHS).

Starting with LHS: [ \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} ]

To combine these fractions, we need a common denominator. The common denominator here is ((1-\sin\theta)(1+\sin\theta)).

[ \frac{(1+\sin\theta) + (1-\sin\theta)}{(1-\sin\theta)(1+\sin\theta)} ]

Simplify the numerator:

[ \frac{1+\sin\theta + 1-\sin\theta}{(1-\sin\theta)(1+\sin\To verify the identity ( \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} = 2 \sec^2(\theta) ), we can start with the left-hand side of the equation and simplify it step by step.

[ \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} ]

To combine these fractions, we find a common denominator:

[ = \frac{(1 + \sin(\theta)) + (1 - \sin(\theta))}{(1 - \sin^2(\theta))} ]

Now, since ( 1 - \sin^2(\theta) = \cos^2(\theta) ), we can simplify further:

[ = \frac{1 + \sin(\theta) + 1 - \sin(\theta)}{\cos^2(\theta)}To verify the identity ( \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} = 2\sec^2\theta ), we can start with the left-hand side (LHS) of the equation and manipulate it until it matches the right-hand side (RHS).

Starting with LHS: [ \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} ]

To combine these fractions, we need a common denominator. The common denominator here is ((1-\sin\theta)(1+\sin\theta)).

[ \frac{(1+\sin\theta) + (1-\sin\theta)}{(1-\sin\theta)(1+\sin\theta)} ]

Simplify the numerator:

[ \frac{1+\sin\theta + 1-\sin\theta}{(1-\sin\theta)(1+\sin\thetaTo verify the identity ( \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} = 2 \sec^2(\theta) ), we can start with the left-hand side of the equation and simplify it step by step.

[ \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} ]

To combine these fractions, we find a common denominator:

[ = \frac{(1 + \sin(\theta)) + (1 - \sin(\theta))}{(1 - \sin^2(\theta))} ]

Now, since ( 1 - \sin^2(\theta) = \cos^2(\theta) ), we can simplify further:

[ = \frac{1 + \sin(\theta) + 1 - \sin(\theta)}{\cos^2(\theta)} \To verify the identity ( \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} = 2\sec^2\theta ), we can start with the left-hand side (LHS) of the equation and manipulate it until it matches the right-hand side (RHS).

Starting with LHS: [ \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} ]

To combine these fractions, we need a common denominator. The common denominator here is ((1-\sin\theta)(1+\sin\theta)).

[ \frac{(1+\sin\theta) + (1-\sin\theta)}{(1-\sin\theta)(1+\sin\theta)} ]

Simplify the numerator:

[ \frac{1+\sin\theta + 1-\sin\theta}{(1-\sin\theta)(1+\sin\theta)}To verify the identity ( \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} = 2 \sec^2(\theta) ), we can start with the left-hand side of the equation and simplify it step by step.

[ \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} ]

To combine these fractions, we find a common denominator:

[ = \frac{(1 + \sin(\theta)) + (1 - \sin(\theta))}{(1 - \sin^2(\theta))} ]

Now, since ( 1 - \sin^2(\theta) = \cos^2(\theta) ), we can simplify further:

[ = \frac{1 + \sin(\theta) + 1 - \sin(\theta)}{\cos^2(\theta)} ]

[To verify the identity ( \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} = 2\sec^2\theta ), we can start with the left-hand side (LHS) of the equation and manipulate it until it matches the right-hand side (RHS).

Starting with LHS: [ \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} ]

To combine these fractions, we need a common denominator. The common denominator here is ((1-\sin\theta)(1+\sin\theta)).

[ \frac{(1+\sin\theta) + (1-\sin\theta)}{(1-\sin\theta)(1+\sin\theta)} ]

Simplify the numerator:

[ \frac{1+\sin\theta + 1-\sin\theta}{(1-\sin\theta)(1+\sin\theta)} \To verify the identity ( \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} = 2 \sec^2(\theta) ), we can start with the left-hand side of the equation and simplify it step by step.

[ \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} ]

To combine these fractions, we find a common denominator:

[ = \frac{(1 + \sin(\theta)) + (1 - \sin(\theta))}{(1 - \sin^2(\theta))} ]

Now, since ( 1 - \sin^2(\theta) = \cos^2(\theta) ), we can simplify further:

[ = \frac{1 + \sin(\theta) + 1 - \sin(\theta)}{\cos^2(\theta)} ]

[ =To verify the identity ( \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} = 2\sec^2\theta ), we can start with the left-hand side (LHS) of the equation and manipulate it until it matches the right-hand side (RHS).

Starting with LHS: [ \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} ]

To combine these fractions, we need a common denominator. The common denominator here is ((1-\sin\theta)(1+\sin\theta)).

[ \frac{(1+\sin\theta) + (1-\sin\theta)}{(1-\sin\theta)(1+\sin\theta)} ]

Simplify the numerator:

[ \frac{1+\sin\theta + 1-\sin\theta}{(1-\sin\theta)(1+\sin\theta)} ] To verify the identity ( \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} = 2 \sec^2(\theta) ), we can start with the left-hand side of the equation and simplify it step by step.

[ \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} ]

To combine these fractions, we find a common denominator:

[ = \frac{(1 + \sin(\theta)) + (1 - \sin(\theta))}{(1 - \sin^2(\theta))} ]

Now, since ( 1 - \sin^2(\theta) = \cos^2(\theta) ), we can simplify further:

[ = \frac{1 + \sin(\theta) + 1 - \sin(\theta)}{\cos^2(\theta)} ]

[ = \To verify the identity ( \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} = 2\sec^2\theta ), we can start with the left-hand side (LHS) of the equation and manipulate it until it matches the right-hand side (RHS).

Starting with LHS: [ \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} ]

To combine these fractions, we need a common denominator. The common denominator here is ((1-\sin\theta)(1+\sin\theta)).

[ \frac{(1+\sin\theta) + (1-\sin\theta)}{(1-\sin\theta)(1+\sin\theta)} ]

Simplify the numerator:

[ \frac{1+\sin\theta + 1-\sin\theta}{(1-\sin\theta)(1+\sin\theta)} ] [ \To verify the identity ( \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} = 2 \sec^2(\theta) ), we can start with the left-hand side of the equation and simplify it step by step.

[ \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} ]

To combine these fractions, we find a common denominator:

[ = \frac{(1 + \sin(\theta)) + (1 - \sin(\theta))}{(1 - \sin^2(\theta))} ]

Now, since ( 1 - \sin^2(\theta) = \cos^2(\theta) ), we can simplify further:

[ = \frac{1 + \sin(\theta) + 1 - \sin(\theta)}{\cos^2(\theta)} ]

[ = \fracTo verify the identity ( \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} = 2\sec^2\theta ), we can start with the left-hand side (LHS) of the equation and manipulate it until it matches the right-hand side (RHS).

Starting with LHS: [ \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} ]

To combine these fractions, we need a common denominator. The common denominator here is ((1-\sin\theta)(1+\sin\theta)).

[ \frac{(1+\sin\theta) + (1-\sin\theta)}{(1-\sin\theta)(1+\sin\theta)} ]

Simplify the numerator:

[ \frac{1+\sin\theta + 1-\sin\theta}{(1-\sin\theta)(1+\sin\theta)} ] [ \frac{To verify the identity ( \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} = 2 \sec^2(\theta) ), we can start with the left-hand side of the equation and simplify it step by step.

[ \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} ]

To combine these fractions, we find a common denominator:

[ = \frac{(1 + \sin(\theta)) + (1 - \sin(\theta))}{(1 - \sin^2(\theta))} ]

Now, since ( 1 - \sin^2(\theta) = \cos^2(\theta) ), we can simplify further:

[ = \frac{1 + \sin(\theta) + 1 - \sin(\theta)}{\cos^2(\theta)} ]

[ = \frac{To verify the identity ( \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} = 2\sec^2\theta ), we can start with the left-hand side (LHS) of the equation and manipulate it until it matches the right-hand side (RHS).

Starting with LHS: [ \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} ]

To combine these fractions, we need a common denominator. The common denominator here is ((1-\sin\theta)(1+\sin\theta)).

[ \frac{(1+\sin\theta) + (1-\sin\theta)}{(1-\sin\theta)(1+\sin\theta)} ]

Simplify the numerator:

[ \frac{1+\sin\theta + 1-\sin\theta}{(1-\sin\theta)(1+\sin\theta)} ] [ \frac{2}{(To verify the identity ( \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} = 2 \sec^2(\theta) ), we can start with the left-hand side of the equation and simplify it step by step.

[ \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} ]

To combine these fractions, we find a common denominator:

[ = \frac{(1 + \sin(\theta)) + (1 - \sin(\theta))}{(1 - \sin^2(\theta))} ]

Now, since ( 1 - \sin^2(\theta) = \cos^2(\theta) ), we can simplify further:

[ = \frac{1 + \sin(\theta) + 1 - \sin(\theta)}{\cos^2(\theta)} ]

[ = \frac{2To verify the identity ( \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} = 2\sec^2\theta ), we can start with the left-hand side (LHS) of the equation and manipulate it until it matches the right-hand side (RHS).

Starting with LHS: [ \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} ]

To combine these fractions, we need a common denominator. The common denominator here is ((1-\sin\theta)(1+\sin\theta)).

[ \frac{(1+\sin\theta) + (1-\sin\theta)}{(1-\sin\theta)(1+\sin\theta)} ]

Simplify the numerator:

[ \frac{1+\sin\theta + 1-\sin\theta}{(1-\sin\theta)(1+\sin\theta)} ] [ \frac{2}{(1-\To verify the identity ( \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} = 2 \sec^2(\theta) ), we can start with the left-hand side of the equation and simplify it step by step.

[ \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} ]

To combine these fractions, we find a common denominator:

[ = \frac{(1 + \sin(\theta)) + (1 - \sin(\theta))}{(1 - \sin^2(\theta))} ]

Now, since ( 1 - \sin^2(\theta) = \cos^2(\theta) ), we can simplify further:

[ = \frac{1 + \sin(\theta) + 1 - \sin(\theta)}{\cos^2(\theta)} ]

[ = \frac{2}{To verify the identity ( \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} = 2\sec^2\theta ), we can start with the left-hand side (LHS) of the equation and manipulate it until it matches the right-hand side (RHS).

Starting with LHS: [ \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} ]

To combine these fractions, we need a common denominator. The common denominator here is ((1-\sin\theta)(1+\sin\theta)).

[ \frac{(1+\sin\theta) + (1-\sin\theta)}{(1-\sin\theta)(1+\sin\theta)} ]

Simplify the numerator:

[ \frac{1+\sin\theta + 1-\sin\theta}{(1-\sin\theta)(1+\sin\theta)} ] [ \frac{2}{(1-\sinTo verify the identity ( \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} = 2 \sec^2(\theta) ), we can start with the left-hand side of the equation and simplify it step by step.

[ \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} ]

To combine these fractions, we find a common denominator:

[ = \frac{(1 + \sin(\theta)) + (1 - \sin(\theta))}{(1 - \sin^2(\theta))} ]

Now, since ( 1 - \sin^2(\theta) = \cos^2(\theta) ), we can simplify further:

[ = \frac{1 + \sin(\theta) + 1 - \sin(\theta)}{\cos^2(\theta)} ]

[ = \frac{2}{\To verify the identity ( \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} = 2\sec^2\theta ), we can start with the left-hand side (LHS) of the equation and manipulate it until it matches the right-hand side (RHS).

Starting with LHS: [ \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} ]

To combine these fractions, we need a common denominator. The common denominator here is ((1-\sin\theta)(1+\sin\theta)).

[ \frac{(1+\sin\theta) + (1-\sin\theta)}{(1-\sin\theta)(1+\sin\theta)} ]

Simplify the numerator:

[ \frac{1+\sin\theta + 1-\sin\theta}{(1-\sin\theta)(1+\sin\theta)} ] [ \frac{2}{(1-\sin\To verify the identity ( \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} = 2 \sec^2(\theta) ), we can start with the left-hand side of the equation and simplify it step by step.

[ \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} ]

To combine these fractions, we find a common denominator:

[ = \frac{(1 + \sin(\theta)) + (1 - \sin(\theta))}{(1 - \sin^2(\theta))} ]

Now, since ( 1 - \sin^2(\theta) = \cos^2(\theta) ), we can simplify further:

[ = \frac{1 + \sin(\theta) + 1 - \sin(\theta)}{\cos^2(\theta)} ]

[ = \frac{2}{\cosTo verify the identity ( \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} = 2\sec^2\theta ), we can start with the left-hand side (LHS) of the equation and manipulate it until it matches the right-hand side (RHS).

Starting with LHS: [ \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} ]

To combine these fractions, we need a common denominator. The common denominator here is ((1-\sin\theta)(1+\sin\theta)).

[ \frac{(1+\sin\theta) + (1-\sin\theta)}{(1-\sin\theta)(1+\sin\theta)} ]

Simplify the numerator:

[ \frac{1+\sin\theta + 1-\sin\theta}{(1-\sin\theta)(1+\sin\theta)} ] [ \frac{2}{(1-\sin\thetaTo verify the identity ( \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} = 2 \sec^2(\theta) ), we can start with the left-hand side of the equation and simplify it step by step.

[ \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} ]

To combine these fractions, we find a common denominator:

[ = \frac{(1 + \sin(\theta)) + (1 - \sin(\theta))}{(1 - \sin^2(\theta))} ]

Now, since ( 1 - \sin^2(\theta) = \cos^2(\theta) ), we can simplify further:

[ = \frac{1 + \sin(\theta) + 1 - \sin(\theta)}{\cos^2(\theta)} ]

[ = \frac{2}{\cos^To verify the identity ( \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} = 2\sec^2\theta ), we can start with the left-hand side (LHS) of the equation and manipulate it until it matches the right-hand side (RHS).

Starting with LHS: [ \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} ]

To combine these fractions, we need a common denominator. The common denominator here is ((1-\sin\theta)(1+\sin\theta)).

[ \frac{(1+\sin\theta) + (1-\sin\theta)}{(1-\sin\theta)(1+\sin\theta)} ]

Simplify the numerator:

[ \frac{1+\sin\theta + 1-\sin\theta}{(1-\sin\theta)(1+\sin\theta)} ] [ \frac{2}{(1-\sin\theta)(To verify the identity ( \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} = 2 \sec^2(\theta) ), we can start with the left-hand side of the equation and simplify it step by step.

[ \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} ]

To combine these fractions, we find a common denominator:

[ = \frac{(1 + \sin(\theta)) + (1 - \sin(\theta))}{(1 - \sin^2(\theta))} ]

Now, since ( 1 - \sin^2(\theta) = \cos^2(\theta) ), we can simplify further:

[ = \frac{1 + \sin(\theta) + 1 - \sin(\theta)}{\cos^2(\theta)} ]

[ = \frac{2}{\cos^2(\To verify the identity ( \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} = 2\sec^2\theta ), we can start with the left-hand side (LHS) of the equation and manipulate it until it matches the right-hand side (RHS).

Starting with LHS: [ \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} ]

To combine these fractions, we need a common denominator. The common denominator here is ((1-\sin\theta)(1+\sin\theta)).

[ \frac{(1+\sin\theta) + (1-\sin\theta)}{(1-\sin\theta)(1+\sin\theta)} ]

Simplify the numerator:

[ \frac{1+\sin\theta + 1-\sin\theta}{(1-\sin\theta)(1+\sin\theta)} ] [ \frac{2}{(1-\sin\theta)(1+\To verify the identity ( \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} = 2 \sec^2(\theta) ), we can start with the left-hand side of the equation and simplify it step by step.

[ \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} ]

To combine these fractions, we find a common denominator:

[ = \frac{(1 + \sin(\theta)) + (1 - \sin(\theta))}{(1 - \sin^2(\theta))} ]

Now, since ( 1 - \sin^2(\theta) = \cos^2(\theta) ), we can simplify further:

[ = \frac{1 + \sin(\theta) + 1 - \sin(\theta)}{\cos^2(\theta)} ]

[ = \frac{2}{\cos^2(\thetaTo verify the identity ( \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} = 2\sec^2\theta ), we can start with the left-hand side (LHS) of the equation and manipulate it until it matches the right-hand side (RHS).

Starting with LHS: [ \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} ]

To combine these fractions, we need a common denominator. The common denominator here is ((1-\sin\theta)(1+\sin\theta)).

[ \frac{(1+\sin\theta) + (1-\sin\theta)}{(1-\sin\theta)(1+\sin\theta)} ]

Simplify the numerator:

[ \frac{1+\sin\theta + 1-\sin\theta}{(1-\sin\theta)(1+\sin\theta)} ] [ \frac{2}{(1-\sin\theta)(1+\sinTo verify the identity ( \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} = 2 \sec^2(\theta) ), we can start with the left-hand side of the equation and simplify it step by step.

[ \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} ]

To combine these fractions, we find a common denominator:

[ = \frac{(1 + \sin(\theta)) + (1 - \sin(\theta))}{(1 - \sin^2(\theta))} ]

Now, since ( 1 - \sin^2(\theta) = \cos^2(\theta) ), we can simplify further:

[ = \frac{1 + \sin(\theta) + 1 - \sin(\theta)}{\cos^2(\theta)} ]

[ = \frac{2}{\cos^2(\theta)}To verify the identity ( \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} = 2\sec^2\theta ), we can start with the left-hand side (LHS) of the equation and manipulate it until it matches the right-hand side (RHS).

Starting with LHS: [ \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} ]

To combine these fractions, we need a common denominator. The common denominator here is ((1-\sin\theta)(1+\sin\theta)).

[ \frac{(1+\sin\theta) + (1-\sin\theta)}{(1-\sin\theta)(1+\sin\theta)} ]

Simplify the numerator:

[ \frac{1+\sin\theta + 1-\sin\theta}{(1-\sin\theta)(1+\sin\theta)} ] [ \frac{2}{(1-\sin\theta)(1+\sin\To verify the identity ( \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} = 2 \sec^2(\theta) ), we can start with the left-hand side of the equation and simplify it step by step.

[ \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} ]

To combine these fractions, we find a common denominator:

[ = \frac{(1 + \sin(\theta)) + (1 - \sin(\theta))}{(1 - \sin^2(\theta))} ]

Now, since ( 1 - \sin^2(\theta) = \cos^2(\theta) ), we can simplify further:

[ = \frac{1 + \sin(\theta) + 1 - \sin(\theta)}{\cos^2(\theta)} ]

[ = \frac{2}{\cos^2(\theta)} \To verify the identity ( \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} = 2\sec^2\theta ), we can start with the left-hand side (LHS) of the equation and manipulate it until it matches the right-hand side (RHS).

Starting with LHS: [ \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} ]

To combine these fractions, we need a common denominator. The common denominator here is ((1-\sin\theta)(1+\sin\theta)).

[ \frac{(1+\sin\theta) + (1-\sin\theta)}{(1-\sin\theta)(1+\sin\theta)} ]

Simplify the numerator:

[ \frac{1+\sin\theta + 1-\sin\theta}{(1-\sin\theta)(1+\sin\theta)} ] [ \frac{2}{(1-\sin\theta)(1+\sin\thetaTo verify the identity ( \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} = 2 \sec^2(\theta) ), we can start with the left-hand side of the equation and simplify it step by step.

[ \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} ]

To combine these fractions, we find a common denominator:

[ = \frac{(1 + \sin(\theta)) + (1 - \sin(\theta))}{(1 - \sin^2(\theta))} ]

Now, since ( 1 - \sin^2(\theta) = \cos^2(\theta) ), we can simplify further:

[ = \frac{1 + \sin(\theta) + 1 - \sin(\theta)}{\cos^2(\theta)} ]

[ = \frac{2}{\cos^2(\theta)} ]

[To verify the identity ( \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} = 2\sec^2\theta ), we can start with the left-hand side (LHS) of the equation and manipulate it until it matches the right-hand side (RHS).

Starting with LHS: [ \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} ]

To combine these fractions, we need a common denominator. The common denominator here is ((1-\sin\theta)(1+\sin\theta)).

[ \frac{(1+\sin\theta) + (1-\sin\theta)}{(1-\sin\theta)(1+\sin\theta)} ]

Simplify the numerator:

[ \frac{1+\sin\theta + 1-\sin\theta}{(1-\sin\theta)(1+\sin\theta)} ] [ \frac{2}{(1-\sin\theta)(1+\sin\theta)}To verify the identity ( \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} = 2 \sec^2(\theta) ), we can start with the left-hand side of the equation and simplify it step by step.

[ \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} ]

To combine these fractions, we find a common denominator:

[ = \frac{(1 + \sin(\theta)) + (1 - \sin(\theta))}{(1 - \sin^2(\theta))} ]

Now, since ( 1 - \sin^2(\theta) = \cos^2(\theta) ), we can simplify further:

[ = \frac{1 + \sin(\theta) + 1 - \sin(\theta)}{\cos^2(\theta)} ]

[ = \frac{2}{\cos^2(\theta)} ]

[ =To verify the identity ( \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} = 2\sec^2\theta ), we can start with the left-hand side (LHS) of the equation and manipulate it until it matches the right-hand side (RHS).

Starting with LHS: [ \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} ]

To combine these fractions, we need a common denominator. The common denominator here is ((1-\sin\theta)(1+\sin\theta)).

[ \frac{(1+\sin\theta) + (1-\sin\theta)}{(1-\sin\theta)(1+\sin\theta)} ]

Simplify the numerator:

[ \frac{1+\sin\theta + 1-\sin\theta}{(1-\sin\theta)(1+\sin\theta)} ] [ \frac{2}{(1-\sin\theta)(1+\sin\theta)} ]

To verify the identity ( \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} = 2 \sec^2(\theta) ), we can start with the left-hand side of the equation and simplify it step by step.

[ \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} ]

To combine these fractions, we find a common denominator:

[ = \frac{(1 + \sin(\theta)) + (1 - \sin(\theta))}{(1 - \sin^2(\theta))} ]

Now, since ( 1 - \sin^2(\theta) = \cos^2(\theta) ), we can simplify further:

[ = \frac{1 + \sin(\theta) + 1 - \sin(\theta)}{\cos^2(\theta)} ]

[ = \frac{2}{\cos^2(\theta)} ]

[ = To verify the identity ( \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} = 2\sec^2\theta ), we can start with the left-hand side (LHS) of the equation and manipulate it until it matches the right-hand side (RHS).

Starting with LHS: [ \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} ]

To combine these fractions, we need a common denominator. The common denominator here is ((1-\sin\theta)(1+\sin\theta)).

[ \frac{(1+\sin\theta) + (1-\sin\theta)}{(1-\sin\theta)(1+\sin\theta)} ]

Simplify the numerator:

[ \frac{1+\sin\theta + 1-\sin\theta}{(1-\sin\theta)(1+\sin\theta)} ] [ \frac{2}{(1-\sin\theta)(1+\sin\theta)} ]

NowTo verify the identity ( \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} = 2 \sec^2(\theta) ), we can start with the left-hand side of the equation and simplify it step by step.

[ \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} ]

To combine these fractions, we find a common denominator:

[ = \frac{(1 + \sin(\theta)) + (1 - \sin(\theta))}{(1 - \sin^2(\theta))} ]

Now, since ( 1 - \sin^2(\theta) = \cos^2(\theta) ), we can simplify further:

[ = \frac{1 + \sin(\theta) + 1 - \sin(\theta)}{\cos^2(\theta)} ]

[ = \frac{2}{\cos^2(\theta)} ]

[ = 2To verify the identity ( \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} = 2\sec^2\theta ), we can start with the left-hand side (LHS) of the equation and manipulate it until it matches the right-hand side (RHS).

Starting with LHS: [ \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} ]

To combine these fractions, we need a common denominator. The common denominator here is ((1-\sin\theta)(1+\sin\theta)).

[ \frac{(1+\sin\theta) + (1-\sin\theta)}{(1-\sin\theta)(1+\sin\theta)} ]

Simplify the numerator:

[ \frac{1+\sin\theta + 1-\sin\theta}{(1-\sin\theta)(1+\sin\theta)} ] [ \frac{2}{(1-\sin\theta)(1+\sin\theta)} ]

Now,To verify the identity ( \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} = 2 \sec^2(\theta) ), we can start with the left-hand side of the equation and simplify it step by step.

[ \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} ]

To combine these fractions, we find a common denominator:

[ = \frac{(1 + \sin(\theta)) + (1 - \sin(\theta))}{(1 - \sin^2(\theta))} ]

Now, since ( 1 - \sin^2(\theta) = \cos^2(\theta) ), we can simplify further:

[ = \frac{1 + \sin(\theta) + 1 - \sin(\theta)}{\cos^2(\theta)} ]

[ = \frac{2}{\cos^2(\theta)} ]

[ = 2 \To verify the identity ( \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} = 2\sec^2\theta ), we can start with the left-hand side (LHS) of the equation and manipulate it until it matches the right-hand side (RHS).

Starting with LHS: [ \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} ]

To combine these fractions, we need a common denominator. The common denominator here is ((1-\sin\theta)(1+\sin\theta)).

[ \frac{(1+\sin\theta) + (1-\sin\theta)}{(1-\sin\theta)(1+\sin\theta)} ]

Simplify the numerator:

[ \frac{1+\sin\theta + 1-\sin\theta}{(1-\sin\theta)(1+\sin\theta)} ] [ \frac{2}{(1-\sin\theta)(1+\sin\theta)} ]

Now, weTo verify the identity ( \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} = 2 \sec^2(\theta) ), we can start with the left-hand side of the equation and simplify it step by step.

[ \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} ]

To combine these fractions, we find a common denominator:

[ = \frac{(1 + \sin(\theta)) + (1 - \sin(\theta))}{(1 - \sin^2(\theta))} ]

Now, since ( 1 - \sin^2(\theta) = \cos^2(\theta) ), we can simplify further:

[ = \frac{1 + \sin(\theta) + 1 - \sin(\theta)}{\cos^2(\theta)} ]

[ = \frac{2}{\cos^2(\theta)} ]

[ = 2 \secTo verify the identity ( \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} = 2\sec^2\theta ), we can start with the left-hand side (LHS) of the equation and manipulate it until it matches the right-hand side (RHS).

Starting with LHS: [ \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} ]

To combine these fractions, we need a common denominator. The common denominator here is ((1-\sin\theta)(1+\sin\theta)).

[ \frac{(1+\sin\theta) + (1-\sin\theta)}{(1-\sin\theta)(1+\sin\theta)} ]

Simplify the numerator:

[ \frac{1+\sin\theta + 1-\sin\theta}{(1-\sin\theta)(1+\sin\theta)} ] [ \frac{2}{(1-\sin\theta)(1+\sin\theta)} ]

Now, we can useTo verify the identity ( \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} = 2 \sec^2(\theta) ), we can start with the left-hand side of the equation and simplify it step by step.

[ \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} ]

To combine these fractions, we find a common denominator:

[ = \frac{(1 + \sin(\theta)) + (1 - \sin(\theta))}{(1 - \sin^2(\theta))} ]

Now, since ( 1 - \sin^2(\theta) = \cos^2(\theta) ), we can simplify further:

[ = \frac{1 + \sin(\theta) + 1 - \sin(\theta)}{\cos^2(\theta)} ]

[ = \frac{2}{\cos^2(\theta)} ]

[ = 2 \sec^To verify the identity ( \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} = 2\sec^2\theta ), we can start with the left-hand side (LHS) of the equation and manipulate it until it matches the right-hand side (RHS).

Starting with LHS: [ \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} ]

To combine these fractions, we need a common denominator. The common denominator here is ((1-\sin\theta)(1+\sin\theta)).

[ \frac{(1+\sin\theta) + (1-\sin\theta)}{(1-\sin\theta)(1+\sin\theta)} ]

Simplify the numerator:

[ \frac{1+\sin\theta + 1-\sin\theta}{(1-\sin\theta)(1+\sin\theta)} ] [ \frac{2}{(1-\sin\theta)(1+\sin\theta)} ]

Now, we can use trigTo verify the identity ( \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} = 2 \sec^2(\theta) ), we can start with the left-hand side of the equation and simplify it step by step.

[ \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} ]

To combine these fractions, we find a common denominator:

[ = \frac{(1 + \sin(\theta)) + (1 - \sin(\theta))}{(1 - \sin^2(\theta))} ]

Now, since ( 1 - \sin^2(\theta) = \cos^2(\theta) ), we can simplify further:

[ = \frac{1 + \sin(\theta) + 1 - \sin(\theta)}{\cos^2(\theta)} ]

[ = \frac{2}{\cos^2(\theta)} ]

[ = 2 \sec^2To verify the identity ( \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} = 2\sec^2\theta ), we can start with the left-hand side (LHS) of the equation and manipulate it until it matches the right-hand side (RHS).

Starting with LHS: [ \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} ]

To combine these fractions, we need a common denominator. The common denominator here is ((1-\sin\theta)(1+\sin\theta)).

[ \frac{(1+\sin\theta) + (1-\sin\theta)}{(1-\sin\theta)(1+\sin\theta)} ]

Simplify the numerator:

[ \frac{1+\sin\theta + 1-\sin\theta}{(1-\sin\theta)(1+\sin\theta)} ] [ \frac{2}{(1-\sin\theta)(1+\sin\theta)} ]

Now, we can use trigonTo verify the identity ( \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} = 2 \sec^2(\theta) ), we can start with the left-hand side of the equation and simplify it step by step.

[ \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} ]

To combine these fractions, we find a common denominator:

[ = \frac{(1 + \sin(\theta)) + (1 - \sin(\theta))}{(1 - \sin^2(\theta))} ]

Now, since ( 1 - \sin^2(\theta) = \cos^2(\theta) ), we can simplify further:

[ = \frac{1 + \sin(\theta) + 1 - \sin(\theta)}{\cos^2(\theta)} ]

[ = \frac{2}{\cos^2(\theta)} ]

[ = 2 \sec^2(\To verify the identity ( \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} = 2\sec^2\theta ), we can start with the left-hand side (LHS) of the equation and manipulate it until it matches the right-hand side (RHS).

Starting with LHS: [ \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} ]

To combine these fractions, we need a common denominator. The common denominator here is ((1-\sin\theta)(1+\sin\theta)).

[ \frac{(1+\sin\theta) + (1-\sin\theta)}{(1-\sin\theta)(1+\sin\theta)} ]

Simplify the numerator:

[ \frac{1+\sin\theta + 1-\sin\theta}{(1-\sin\theta)(1+\sin\theta)} ] [ \frac{2}{(1-\sin\theta)(1+\sin\theta)} ]

Now, we can use trigonometricTo verify the identity ( \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} = 2 \sec^2(\theta) ), we can start with the left-hand side of the equation and simplify it step by step.

[ \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} ]

To combine these fractions, we find a common denominator:

[ = \frac{(1 + \sin(\theta)) + (1 - \sin(\theta))}{(1 - \sin^2(\theta))} ]

Now, since ( 1 - \sin^2(\theta) = \cos^2(\theta) ), we can simplify further:

[ = \frac{1 + \sin(\theta) + 1 - \sin(\theta)}{\cos^2(\theta)} ]

[ = \frac{2}{\cos^2(\theta)} ]

[ = 2 \sec^2(\thetaTo verify the identity ( \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} = 2\sec^2\theta ), we can start with the left-hand side (LHS) of the equation and manipulate it until it matches the right-hand side (RHS).

Starting with LHS: [ \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} ]

To combine these fractions, we need a common denominator. The common denominator here is ((1-\sin\theta)(1+\sin\theta)).

[ \frac{(1+\sin\theta) + (1-\sin\theta)}{(1-\sin\theta)(1+\sin\theta)} ]

Simplify the numerator:

[ \frac{1+\sin\theta + 1-\sin\theta}{(1-\sin\theta)(1+\sin\theta)} ] [ \frac{2}{(1-\sin\theta)(1+\sin\theta)} ]

Now, we can use trigonometric identitiesTo verify the identity ( \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} = 2 \sec^2(\theta) ), we can start with the left-hand side of the equation and simplify it step by step.

[ \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} ]

To combine these fractions, we find a common denominator:

[ = \frac{(1 + \sin(\theta)) + (1 - \sin(\theta))}{(1 - \sin^2(\theta))} ]

Now, since ( 1 - \sin^2(\theta) = \cos^2(\theta) ), we can simplify further:

[ = \frac{1 + \sin(\theta) + 1 - \sin(\theta)}{\cos^2(\theta)} ]

[ = \frac{2}{\cos^2(\theta)} ]

[ = 2 \sec^2(\theta) \To verify the identity ( \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} = 2\sec^2\theta ), we can start with the left-hand side (LHS) of the equation and manipulate it until it matches the right-hand side (RHS).

Starting with LHS: [ \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} ]

To combine these fractions, we need a common denominator. The common denominator here is ((1-\sin\theta)(1+\sin\theta)).

[ \frac{(1+\sin\theta) + (1-\sin\theta)}{(1-\sin\theta)(1+\sin\theta)} ]

Simplify the numerator:

[ \frac{1+\sin\theta + 1-\sin\theta}{(1-\sin\theta)(1+\sin\theta)} ] [ \frac{2}{(1-\sin\theta)(1+\sin\theta)} ]

Now, we can use trigonometric identities to simplifyTo verify the identity ( \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} = 2 \sec^2(\theta) ), we can start with the left-hand side of the equation and simplify it step by step.

[ \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} ]

To combine these fractions, we find a common denominator:

[ = \frac{(1 + \sin(\theta)) + (1 - \sin(\theta))}{(1 - \sin^2(\theta))} ]

Now, since ( 1 - \sin^2(\theta) = \cos^2(\theta) ), we can simplify further:

[ = \frac{1 + \sin(\theta) + 1 - \sin(\theta)}{\cos^2(\theta)} ]

[ = \frac{2}{\cos^2(\theta)} ]

[ = 2 \sec^2(\theta) ]

To verify the identity ( \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} = 2\sec^2\theta ), we can start with the left-hand side (LHS) of the equation and manipulate it until it matches the right-hand side (RHS).

Starting with LHS: [ \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} ]

To combine these fractions, we need a common denominator. The common denominator here is ((1-\sin\theta)(1+\sin\theta)).

[ \frac{(1+\sin\theta) + (1-\sin\theta)}{(1-\sin\theta)(1+\sin\theta)} ]

Simplify the numerator:

[ \frac{1+\sin\theta + 1-\sin\theta}{(1-\sin\theta)(1+\sin\theta)} ] [ \frac{2}{(1-\sin\theta)(1+\sin\theta)} ]

Now, we can use trigonometric identities to simplify (To verify the identity ( \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} = 2 \sec^2(\theta) ), we can start with the left-hand side of the equation and simplify it step by step.

[ \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} ]

To combine these fractions, we find a common denominator:

[ = \frac{(1 + \sin(\theta)) + (1 - \sin(\theta))}{(1 - \sin^2(\theta))} ]

Now, since ( 1 - \sin^2(\theta) = \cos^2(\theta) ), we can simplify further:

[ = \frac{1 + \sin(\theta) + 1 - \sin(\theta)}{\cos^2(\theta)} ]

[ = \frac{2}{\cos^2(\theta)} ]

[ = 2 \sec^2(\theta) ]

Thus,To verify the identity ( \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} = 2\sec^2\theta ), we can start with the left-hand side (LHS) of the equation and manipulate it until it matches the right-hand side (RHS).

Starting with LHS: [ \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} ]

To combine these fractions, we need a common denominator. The common denominator here is ((1-\sin\theta)(1+\sin\theta)).

[ \frac{(1+\sin\theta) + (1-\sin\theta)}{(1-\sin\theta)(1+\sin\theta)} ]

Simplify the numerator:

[ \frac{1+\sin\theta + 1-\sin\theta}{(1-\sin\theta)(1+\sin\theta)} ] [ \frac{2}{(1-\sin\theta)(1+\sin\theta)} ]

Now, we can use trigonometric identities to simplify ( (To verify the identity ( \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} = 2 \sec^2(\theta) ), we can start with the left-hand side of the equation and simplify it step by step.

[ \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} ]

To combine these fractions, we find a common denominator:

[ = \frac{(1 + \sin(\theta)) + (1 - \sin(\theta))}{(1 - \sin^2(\theta))} ]

Now, since ( 1 - \sin^2(\theta) = \cos^2(\theta) ), we can simplify further:

[ = \frac{1 + \sin(\theta) + 1 - \sin(\theta)}{\cos^2(\theta)} ]

[ = \frac{2}{\cos^2(\theta)} ]

[ = 2 \sec^2(\theta) ]

Thus, theTo verify the identity ( \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} = 2\sec^2\theta ), we can start with the left-hand side (LHS) of the equation and manipulate it until it matches the right-hand side (RHS).

Starting with LHS: [ \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} ]

To combine these fractions, we need a common denominator. The common denominator here is ((1-\sin\theta)(1+\sin\theta)).

[ \frac{(1+\sin\theta) + (1-\sin\theta)}{(1-\sin\theta)(1+\sin\theta)} ]

Simplify the numerator:

[ \frac{1+\sin\theta + 1-\sin\theta}{(1-\sin\theta)(1+\sin\theta)} ] [ \frac{2}{(1-\sin\theta)(1+\sin\theta)} ]

Now, we can use trigonometric identities to simplify ( (1To verify the identity ( \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} = 2 \sec^2(\theta) ), we can start with the left-hand side of the equation and simplify it step by step.

[ \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} ]

To combine these fractions, we find a common denominator:

[ = \frac{(1 + \sin(\theta)) + (1 - \sin(\theta))}{(1 - \sin^2(\theta))} ]

Now, since ( 1 - \sin^2(\theta) = \cos^2(\theta) ), we can simplify further:

[ = \frac{1 + \sin(\theta) + 1 - \sin(\theta)}{\cos^2(\theta)} ]

[ = \frac{2}{\cos^2(\theta)} ]

[ = 2 \sec^2(\theta) ]

Thus, the identityTo verify the identity ( \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} = 2\sec^2\theta ), we can start with the left-hand side (LHS) of the equation and manipulate it until it matches the right-hand side (RHS).

Starting with LHS: [ \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} ]

To combine these fractions, we need a common denominator. The common denominator here is ((1-\sin\theta)(1+\sin\theta)).

[ \frac{(1+\sin\theta) + (1-\sin\theta)}{(1-\sin\theta)(1+\sin\theta)} ]

Simplify the numerator:

[ \frac{1+\sin\theta + 1-\sin\theta}{(1-\sin\theta)(1+\sin\theta)} ] [ \frac{2}{(1-\sin\theta)(1+\sin\theta)} ]

Now, we can use trigonometric identities to simplify ( (1-\To verify the identity ( \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} = 2 \sec^2(\theta) ), we can start with the left-hand side of the equation and simplify it step by step.

[ \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} ]

To combine these fractions, we find a common denominator:

[ = \frac{(1 + \sin(\theta)) + (1 - \sin(\theta))}{(1 - \sin^2(\theta))} ]

Now, since ( 1 - \sin^2(\theta) = \cos^2(\theta) ), we can simplify further:

[ = \frac{1 + \sin(\theta) + 1 - \sin(\theta)}{\cos^2(\theta)} ]

[ = \frac{2}{\cos^2(\theta)} ]

[ = 2 \sec^2(\theta) ]

Thus, the identity is verifiedTo verify the identity ( \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} = 2\sec^2\theta ), we can start with the left-hand side (LHS) of the equation and manipulate it until it matches the right-hand side (RHS).

Starting with LHS: [ \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} ]

To combine these fractions, we need a common denominator. The common denominator here is ((1-\sin\theta)(1+\sin\theta)).

[ \frac{(1+\sin\theta) + (1-\sin\theta)}{(1-\sin\theta)(1+\sin\theta)} ]

Simplify the numerator:

[ \frac{1+\sin\theta + 1-\sin\theta}{(1-\sin\theta)(1+\sin\theta)} ] [ \frac{2}{(1-\sin\theta)(1+\sin\theta)} ]

Now, we can use trigonometric identities to simplify ( (1-\sin\To verify the identity ( \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} = 2 \sec^2(\theta) ), we can start with the left-hand side of the equation and simplify it step by step.

[ \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} ]

To combine these fractions, we find a common denominator:

[ = \frac{(1 + \sin(\theta)) + (1 - \sin(\theta))}{(1 - \sin^2(\theta))} ]

Now, since ( 1 - \sin^2(\theta) = \cos^2(\theta) ), we can simplify further:

[ = \frac{1 + \sin(\theta) + 1 - \sin(\theta)}{\cos^2(\theta)} ]

[ = \frac{2}{\cos^2(\theta)} ]

[ = 2 \sec^2(\theta) ]

Thus, the identity is verified.To verify the identity ( \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} = 2\sec^2\theta ), we can start with the left-hand side (LHS) of the equation and manipulate it until it matches the right-hand side (RHS).

Starting with LHS: [ \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} ]

To combine these fractions, we need a common denominator. The common denominator here is ((1-\sin\theta)(1+\sin\theta)).

[ \frac{(1+\sin\theta) + (1-\sin\theta)}{(1-\sin\theta)(1+\sin\theta)} ]

Simplify the numerator:

[ \frac{1+\sin\theta + 1-\sin\theta}{(1-\sin\theta)(1+\sin\theta)} ] [ \frac{2}{(1-\sin\theta)(1+\sin\theta)} ]

Now, we can use trigonometric identities to simplify ( (1-\sin\thetaTo verify the identity ( \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} = 2 \sec^2(\theta) ), we can start with the left-hand side of the equation and simplify it step by step.

[ \frac{1}{1 - \sin(\theta)} + \frac{1}{1 + \sin(\theta)} ]

To combine these fractions, we find a common denominator:

[ = \frac{(1 + \sin(\theta)) + (1 - \sin(\theta))}{(1 - \sin^2(\theta))} ]

Now, since ( 1 - \sin^2(\theta) = \cos^2(\theta) ), we can simplify further:

[ = \frac{1 + \sin(\theta) + 1 - \sin(\theta)}{\cos^2(\theta)} ]

[ = \frac{2}{\cos^2(\theta)} ]

[ = 2 \sec^2(\theta) ]

Thus, the identity is verified.To verify the identity ( \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} = 2\sec^2\theta ), we can start with the left-hand side (LHS) of the equation and manipulate it until it matches the right-hand side (RHS).

Starting with LHS: [ \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} ]

To combine these fractions, we need a common denominator. The common denominator here is ((1-\sin\theta)(1+\sin\theta)).

[ \frac{(1+\sin\theta) + (1-\sin\theta)}{(1-\sin\theta)(1+\sin\theta)} ]

Simplify the numerator:

[ \frac{1+\sin\theta + 1-\sin\theta}{(1-\sin\theta)(1+\sin\theta)} ] [ \frac{2}{(1-\sin\theta)(1+\sin\theta)} ]

Now, we can use trigonometric identities to simplify ( (1-\sin\theta)(1+\sin\theta) ).

[ (1-\sin\theta)(1+\sin\theta) = 1 - \sin^2\theta ]

Using the Pythagorean identity ( \sin^2\theta + \cos^2\theta = 1 ), we have ( \sin^2\theta = 1 - \cos^2\theta ).

Substitute this into the expression:

[ 1 - \sin^2\theta = 1 - (1 - \cos^2\theta) = \cos^2\theta ]

So, our expression becomes:

[ \frac{2}{\cos^2\theta} ]

And since (\sec\theta = \frac{1}{\cos\theta}), we can rewrite the expression as:

[ 2\sec^2\theta ]

Thus, we have verified the identity ( \frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} = 2\sec^2\theta ).