Question

Prove that ${\cos }^{4}2x=\frac{1+{\cos }^{2}4x+2\cos 4x}{4}$

Answer 1

We know that,

$\cos 4x=2co{s}^{2}2x-1$

$2{\cos }^{2}2x=1+\cos 4x$

${\cos }^{2}2x=\frac{1+\cos 4x}{2}$

Squaring both side and we get,

${\left({\cos }^{2}2x\right)}^2={\left(\frac{1+\cos 4x}{2}\right)}^2$

${\cos }^{4}2x=\frac{1+{\cos }^{2}4x+2\cos 4x}{4}$