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