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