## Trigonometry 7th Edition

As left side transforms into right side, hence given identity- $\sin^{4} A - \cos^{4} A$ = $1 - 2 \cos^{2} A$ is true.
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.