Question

Prove that:
$\sqrt{2+\sqrt{2+2\cos 4x}}=2\cos x$

Answer 1

$$\begin{gathered} \mathrm{LHS}=\sqrt{2+\sqrt{2+2\cos 4x}} \\ =\sqrt{2+\sqrt{2\left(1+\cos 4x\right)}} \\ =\sqrt{2+\sqrt{2\times 2{\cos }^{2}2x}}\left(\because 2{\cos }^{2}2x=1+\cos 4x\right) \\ =\sqrt{2+2\cos 2\mathrm{x}} \end{gathered}$$

$$\begin{gathered} =\sqrt{2\left(1+\cos 2x\right)} \\ =\sqrt{2.2{\cos }^{2}x}\left(\because 2{\cos }^{2}x=1+\cos 2x\right) \\ =2\cos x=\mathrm{RHS} \\ \mathrm{Hence}\mathrm{proved}. \end{gathered}$$