If

Given for $\frac{\pi}{4}

$\sqrt{2+\sqrt{2+2 \cos 4 x}}$

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

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

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

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

$\left\{\begin{array}{l}\text { Since } \frac{\pi}{4}

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

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

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

$=2|\sin x|$

$\sqrt{2+\sqrt{2+2 \cos 4 x}}=2 \sin x \quad\left(\because\right.$ In $\frac{\pi}{4}