Verify each identity \frac{ \cos^(2)x }{ \csc{}^(2)x } = ( \sin ^(2)x )/( \sec ^(2)x )

Question

verify each identity
`\frac{ \cos^(2)x }{ \csc{}^(2)x } = ( \sin ^(2)x )/( \sec ^(2)x )`

Answer 1

Verified

Step-by-step explanation:

Given the identity:

`\frac{ \cos^(2)x }{ \csc{}^(2)x }=( \sin^(2)x )/( \sec^(2)x )`

We apply the following identity:

`\begin{gathered} \csc =(1)/(\sin)\implies\sin =(1)/(\csc ) \\ \sec =(1)/(\cos)\implies\cos =(1)/(\sec ) \end{gathered}`

Therefore:

`\begin{gathered} \frac{\cos^2x}{\csc{}^2x}\stackrel{?}{=}(\sin^2x)/(\sec^2x) \\ \cos ^2x*\frac{1}{\csc{}^2x}\stackrel{?}{=}\sin ^2x*(1)/(\sec ^2x) \\ \cos ^2x*\sin ^2x\stackrel{?}{=}\sin ^2x*\cos ^2x \\ \implies\sin ^2x\cos ^2x=\sin ^2x\cos ^2x \end{gathered}`

Thus, the identity is verified since both sides of the identity are equal.