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