Question

Verify which of the following are identities.

Answer 1

Only the second equation is an identity

Explanation:

`$8 (\tan^2(\theta))/(\sec(\theta)) \csc^2(\theta)=8 \csc(\theta)$`

Note that

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

You can confirm it:

`$(\sin^2(\theta))/(\cos^2(\theta))\cdot (1)/(\sin^2(\theta))= (1)/(cos^2(\theta))= \sec^2(\theta)$`

Therefore

`$8 (\sec^2(\theta))/(\sec(\theta)) =8 \csc(\theta)$`

`$8 (\sec(\theta))/(1) =8 \csc(\theta)$`

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

It is not an Identity

Let's the second one

`$13 (\tan^2(\theta))/(\sec(\theta)) \csc^2(\theta)=13\sec(\theta)$`

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