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