please help with the following question ....?

Dear Student
Put x in place of theta.
R.H.S = 
$$\begin{gathered} \sqrt{2+\sqrt{2+2\cos 4\theta }} \\ =\sqrt{2+\sqrt{2+2\left(2{\cos }^{2}2\theta -1\right)}} \\ =\sqrt{2+\sqrt{2+4{\cos }^{2}2\theta -2}} \\ =\sqrt{2+\sqrt{2^2{\cos }^{2}2\theta }} \\ =\sqrt{2+2\cos 2\theta } \\ =\sqrt{2+2\left(2{\cos }^2\theta -1\right)} \\ =2\cos \theta \end{gathered}$$
= L.H.S