Question

Prove the following identity:
$1-{\sin }^4\alpha -{\cos }^4\alpha =\frac{1}{2}{\sin }^{2}2\alpha$

Answer 1

$1-({\sin }^4\alpha +{\cos }^4\alpha )$

$=1-({\sin }^2\alpha +{\cos }^2\alpha -{2\sin }^2\alpha {\cos }^2\alpha )$

$=1-\{1-\frac{{\left(2\sin \alpha \cos \alpha \right)}^2}{2}\}$

$=\frac{1}{2}{\sin }^{2}2\alpha$