Question

prove that
${\cos }^4\theta -{\sin }^4\theta =1-2{\sin }^2\theta$

Answer 1

${\cos }^4\theta -{\sin }^4\theta$

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

$[\because {\cos }^2\theta +{\sin }^2\theta =1]$

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

$=cos2\theta$

$[\because {\cos }^2\theta -{\sin }^2\theta =cos2\theta ]$

$=1-2{\sin }^2\theta$

$[\because 1-2{\sin }^2\theta =\cos 2\theta ]$

hence proven.