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