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