Question

Prove that ${\cos }^4\theta -{\sin }^4\theta =\cos 2\theta$.

Answer 1

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

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

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

$=\cos 2\theta$

$\Rightarrow$ LHS $=$ RHS

Hence proved