prove that $\underbrace{\sqrt{2+\sqrt{2+\sqrt{2+\dots+\sqrt{2}}}}}_{\text{n radicals}}=2\cos{\frac{\pi}{2^{n+1}}}$
Proof of an equality involving cosine $\sqrt{2 + \sqrt{2 + \cdots + \sqrt{2 + \sqrt{2}}}}\ =\ 2\cos (\pi/2^{n+1})$