Answer
As right side transforms into left side, hence given identity-
$1 - \sin\theta$ = $\frac{\cos^{2}\theta}{1 + \sin \theta}$ is true.
Work Step by Step
Given identity is-
$1 - \sin\theta$ = $\frac{\cos^{2}\theta}{1 + \sin \theta}$
Taking R.S.
$\frac{\cos^{2}\theta}{1 + \sin \theta}$
= $\frac{\cos^{2}\theta}{1 + \sin \theta}. \frac{1 - \sin \theta}{1 - \sin \theta}$
{Multiplying the numerator and denominator by the conjugate of the denominator, $(1 -\sin \theta)$}
= $\frac{\cos^{2}\theta (1 - \sin \theta)}{(1 + \sin \theta) (1 - \sin \theta)}$
= $\frac{\cos^{2}\theta (1 - \sin \theta)}{1 - \sin^{2}\theta}$
{Recall $(a+b)(a-b) $ = $a^{2} - b^{2}$ }
= $\frac{\cos^{2}\theta (1 - \sin \theta)}{\cos^{2}\theta}$
( From first Pythagorean identity, $1 - \sin^{2}\theta$ = $\cos^{2}\theta$)
= $1 - \sin\theta$
= L.S.
As right side transforms into left side, hence given identity-
$1 - \sin\theta$ = $\frac{\cos^{2}\theta}{1 + \sin \theta}$ is true.
