Answer
Showed that given statement, $\csc\theta -\sin\theta$ = $\frac{\cos^{2}\theta}{\sin\theta}$,
is an identity as left side transforms into right side.
Work Step by Step
Given statement is-
$\csc\theta -\sin\theta$ = $\frac{\cos^{2}\theta}{\sin\theta}$
Left Side = $\csc\theta -\sin\theta$
= $\frac{1}{\sin\theta} -\sin\theta$
( Using reciprocal identity for $\csc\theta$)
= $\frac{1}{\sin\theta} - \sin\theta\times\frac{\sin\theta}{\sin\theta}$
= $\frac{1}{\sin\theta} - \frac{\sin^{2}\theta}{\sin\theta}$
= $\frac{1-\sin^{2}\theta}{\sin\theta}$
= $\frac{\cos^{2}\theta}{\sin\theta}$
[ From first Pythagorean identity, $ (1 - \sin^{2}\theta)$ can be written as $\cos^{2}\theta$]
= Right Side
i.e. Left Side transforms into Right Side
i.e. Given statement, $\csc\theta -\sin\theta$ = $\frac{\cos^{2}\theta}{\sin\theta}$,
is an identity.
