Question:
Prove that:
$\sqrt{2+\sqrt{2+2 \cos 4 x}}=2 \cos x$
Solution:
LHS $=\sqrt{2+\sqrt{2+2 \cos 4 x}}$
$=\sqrt{2+\sqrt{2(1+\cos 4 x)}}$
$=\sqrt{2+\sqrt{2 \times 2 \cos ^{2} 2 x}} \quad\left(\because 2 \cos ^{2} 2 x=1+\cos 4 x\right)$
$=\sqrt{2+2 \cos 2 x}$
$=\sqrt{2(1+\cos 2 x)}$
$=\sqrt{2.2 \cos ^{2} x} \quad\left(\because 2 \cos ^{2} x=1+\cos 2 x\right)$
$=2 \cos x=\mathrm{RHS}$
Hence proved.
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