Answer
As left side transforms into right side, hence given identity-
$\cot\theta \cos \theta + \sin\theta$ = $\csc\theta$ is true.
Work Step by Step
Given identity is-
$\cot\theta \cos \theta + \sin\theta$ = $\csc\theta$
Taking L.S.
$\cot\theta \cos \theta + \sin\theta$
= $\frac{\cos\theta}{\sin \theta} . \cos\theta + \sin\theta . \frac{\sin\theta}{\sin\theta}$
( Using ratio identity for $\cot\theta$)
= $\frac{\cos^{2}\theta}{\sin \theta} + \frac{\sin^{2}\theta}{\sin\theta}$
= $\frac{\cos^{2}\theta + \sin^{2}\theta }{\sin \theta} $
= $\frac{1 }{\sin \theta} $
( Recall first Pythagorean identity, $\cos^{2}\theta + \sin^{2}\theta$ = 1)
= $\csc\theta$
= R.S.
As left side transforms into right side, hence given identity-
$\cot\theta \cos \theta + \sin\theta$ = $\csc\theta$ is true.
