Question

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

Answer 1

$2({\cos }^{4}x+{\sin }^{4}x)$

R.H.S$=2({({\cos }^{2}x+{\sin }^{2}x)}^2-2{\sin }^{2}x{\cos }^{2}x)$

$=2(1-\frac{1}{2}{\sin }^{2}2x)$

$=2(1-\frac{1}{2}(1-{\cos }^{2}2x))$

$=1+{\cos }^{2}2x$

$=$L.H.S

Hence proved.