# Prove that sec(2theta)=sec^2(theta) divided by (2-sec^2(theta))?

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Prove #sec2theta=sec^2theta/(2-sec^2theta)#

Prove

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To prove the identity ( \sec(2\theta) = \frac{\sec^2(\theta)}{2 - \sec^2(\theta)} ), we can start with the left-hand side (LHS) and manipulate it to match the right-hand side (RHS).

Starting with LHS:

[ \sec(2\theta) ]

Using the double angle identity for secant:

[ \sec(2\theta) = \frac{1}{\cos(2\theta)} ]

Using the double angle identity for cosine:

[ \cos(2\theta) = \cos^2(\theta) - \sin^2(\theta) ]

[ \cos(2\theta) = (1 - \sin^2(\theta)) - \sin^2(\theta) ]

[ \cos(2\theta) = 1 - 2\sin^2(\theta) ]

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

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

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

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

This matches the right-hand side (RHS), thus proving the identity.

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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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