Question

Verify which of the following are identities.

Answer 1

Answer:

Only the second equation is an identity

Step-by-step explanation:

$8 \frac{\tan^2(\theta)}{\sec(\theta)} \csc^2(\theta)=8 \csc(\theta)$

Note that

$\tan^2(\theta)\csc^2(\theta)=\sec^2(\theta)$

You can confirm it:

$\frac{\sin^2(\theta)}{\cos^2(\theta)}\cdot \frac{1}{\sin^2(\theta)}= \frac{1}{cos^2(\theta)}= \sec^2(\theta)$

Therefore

$8 \frac{\sec^2(\theta)}{\sec(\theta)} =8 \csc(\theta)$

$8 \frac{\sec(\theta)}{1} =8 \csc(\theta)$

$8\sec(\theta)=8\csc(\theta)$

It is not an Identity

Let's the second one

$13 \frac{\tan^2(\theta)}{\sec(\theta)} \csc^2(\theta)=13\sec(\theta)$

In this case, we already performed the calculations, so it is true. It is an Identity.