# How do you prove #(cscx + cotx)^2 = (1 + cosx) / (1 - cosx)#?

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To prove the trigonometric identity ( (\csc x + \cot x)^2 = \frac{1 + \cos x}{1 - \cos x} ), follow these steps:

Start with the left-hand side (LHS) of the equation: [ (\csc x + \cot x)^2 ]

Expand and simplify the expression: [ (\csc x + \cot x)^2 = \csc^2 x + 2\csc x \cot x + \cot^2 x ]

Use the trigonometric identities: [ \csc^2 x = \frac{1}{\sin^2 x} ] [ \cot x = \frac{\cos x}{\sin x} ] [ \cot^2 x = \left( \frac{\cos x}{\sin x} \right)^2 = \frac{\cos^2 x}{\sin^2 x} ]

Substitute these identities into the expression: [ \frac{1}{\sin^2 x} + 2\frac{\cos x}{\sin x} \cdot \frac{\cos x}{\sin x} + \frac{\cos^2 x}{\sin^2 x} ]

Combine the fractions: [ \frac{1}{\sin^2 x} + \frac{2\cos^2 x}{\sin^2 x} + \frac{\cos^2 x}{\sin^2 x} ]

Combine the terms: [ \frac{1 + 3\cos^2 x}{\sin^2 x} ]

Now, use the Pythagorean identity ( 1 = \sin^2 x + \cos^2 x ) to simplify further: [ 1 + 3\cos^2 x = \sin^2 x + 3\cos^2 x = \sin^2 x + 3(1 - \sin^2 x) ] [ = \sin^2 x + 3 - 3\sin^2 x = 3 - 2\sin^2 x ]

Substitute this back into the expression: [ \frac{3 - 2\sin^2 x}{\sin^2 x} = \frac{3 - 2(1 - \cos^2 x)}{1 - \cos^2 x} ] [ = \frac{3 - 2 + 2\cos^2 x}{1 - \cos^2 x} = \frac{1 + 2\cos^2 x}{1 - \cos^2 x} ]

Since ( 1 - \cos^2 x = \sin^2 x ), the expression simplifies to: [ \frac{1 + \cos x}{1 - \cos x} ]

This is the same as the right-hand side (RHS) of the equation, so the identity is proven.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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