If

We have,

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

$=\sqrt{2+\sqrt{2.2 \cos ^{2} x}}$

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

$=\sqrt{2-2 \cos x} \quad\left(\because \frac{\pi}{2}

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

$=\sqrt{2.2 \sin ^{2} \frac{x}{2}}$

$=2\left|\sin \frac{x}{2}\right|$

$=2 \sin \frac{x}{2} \quad\left(\because \frac{\pi}{4}<\frac{x}{2}<\frac{\pi}{2}\right)$