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