Answer
As left side transforms into right side, hence given identity-
$\frac{1 - \cos^{4} \theta}{1 + \cos^{2} \theta}$ = $\sin^{2} \theta$ is true.
Work Step by Step
Given identity is-
$\frac{1 - \cos^{4} \theta}{1 + \cos^{2} \theta}$ = $\sin^{2} \theta$
Taking L.S.
$\frac{1 - \cos^{4} \theta}{1 + \cos^{2} \theta}$
= $\frac{(1)^{2} - (\cos^{2} \theta)^{2}}{1 + \cos^{2} \theta}$
= $\frac{(1 + \cos^{2} \theta). (1 - \cos^{2} \theta)}{1 + \cos^{2} \theta}$
{Recall $a^{2} - b^{2}$ = (a+b)(a-b)}
= $1 - \cos^{2} \theta$
= $\sin^{2}\theta$
( Using first Pythagorean identity, $1 -\cos^{2} \theta = \sin^{2} \theta$)
= R.S.
As left side transforms into right side, hence given identity-
$\frac{1 - \cos^{4} \theta}{1 + \cos^{2} \theta}$ = $\sin^{2} \theta$ is true.
