Question

verify each identity $\frac{ \cos^{2}x }{ \csc{}^{2}x } = \frac{ \sin ^{2}x }{ \sec ^{2}x }$

Answer 1

Answer:

Verified

Explanation:

Given the identity:

$\frac{ \cos^{2}x }{ \csc{}^{2}x }=\frac{ \sin^{2}x }{ \sec^{2}x }$

We apply the following identity:

$\begin{gathered} \csc =\frac{1}{\sin}\implies\sin =\frac{1}{\csc } \\ \sec =\frac{1}{\cos}\implies\cos =\frac{1}{\sec } \end{gathered}$

Therefore:

$\begin{gathered} \frac{\cos^2x}{\csc{}^2x}\stackrel{?}{=}\frac{\sin^2x}{\sec^2x} \\ \cos ^2x\times\frac{1}{\csc{}^2x}\stackrel{?}{=}\sin ^2x\times\frac{1}{\sec ^2x} \\ \cos ^2x\times\sin ^2x\stackrel{?}{=}\sin ^2x\times\cos ^2x \\ \implies\sin ^2x\cos ^2x=\sin ^2x\cos ^2x \end{gathered}$

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