Question

Prove that:
$1+{\cos }^{2}2x=2({\cos }^{4}x+{\sin }^{4}x)$

Answer 1

$RHS=2\left({\cos }^{4}x+{\sin }^{4}x\right)$

$=2\left({\left({\cos }^{2}x\right)}^2+{\left({\sin }^{2}x\right)}^2\right)$

$=2\left(\left({\cos }^{2}x+{\sin }^{2}x\right)-2{\sin }^{2}x{\cos }^{2}x\right)$

$=2-4{\sin }^{2}x{\cos }^{2}x$

$=2-4.\frac{1-\cos 2x}{2}\times \frac{1+\cos 2x}{2}$

$=2-\left(1-{\cos }^{2}2x\right)=1+{\cos }^{2}2x$

Hence Proved.