Question

Prove the following identity:
$\sqrt{2+\sqrt{2+2\cos 4x}}=2\cos x,0<x<\frac{\pi }{4}$

Answer 1

To prove: $\sqrt{2+\sqrt{2+2\cos 4x}}=2\cos x,0<x<\frac{\pi }{4}$

L.H.S.
$=\sqrt{2+\sqrt{2+2\cos 4x}}$

$=\sqrt{2+\sqrt{2(1+\cos 4x)}}$

$=\sqrt{2+\sqrt{2\times 2{\cos }^{2}2x}}(\because {\cos }^{2}2x=\frac{1+\cos 4x}{2})$

$=\sqrt{2+2\cos 2x}$

$=\sqrt{2(1+\cos 2x)}$

$=\sqrt{2\times 2{\cos }^{2}x}(\because {\cos }^{2}x=\frac{1+\cos 2x}{2})$

$=2\cos x$
=R.H.S.
Hence proved.