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