Question

Prove that:

$\sqrt{2+\sqrt{2+2\cos 4\theta }}=2\cos \theta$

Answer 1

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

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