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