Question

Please help me with these two questions so i know i did it rightverify each identity

Answer 1

Given the identity:

`(sec^2x)/(sec^2x-1)=\csc ^2x`

Reall from trigonometric identities:

`\text{sec}^2x-1=\tan ^2x`

Therefore:

`\begin{gathered} (sec^2x)/(sec^2x-1)=(sec^2x)/(\tan^2x) \\ \implies(sec^2x)/(\tan^2x)=(1)/(\cos^2x)/(\sin^2x)/(\cos^2x)\text{ where }\begin{cases}\sec ^2x=(1)/(\cos^2x) \\ \tan ^2x=(\sin^2x)/(\cos^2x)\end{cases} \end{gathered}`

Change the division sign to multiplication.

`\begin{gathered} (1)/(\cos^2x)*(\cos^2x)/(\sin^2x)=(1)/(\sin ^2x) \\ \text{The inverse of sine is cosecant.} \\ \implies(sec^2x)/(sec^2x-1)=\csc ^2x \end{gathered}`

Thus, the identity is proved.