value of $a,b$ in trigonometric expression

Question

If $a,b$ are positive integers such that $\displaystyle \sqrt{8+\sqrt{32+\sqrt{768}}}=a\cos\bigg(\frac{\pi}{b}\bigg)$

what i try: i am trying to convert it into $\displaystyle \sqrt{2+2\cos 8x}=\sqrt{2}\cos 4x$

How do i solve it Help me please

Answer 1

You're on the right track. The surd is $$2\sqrt{2(1+\sqrt{2+\sqrt{3}})}=2\sqrt{2(1+\sqrt{2(1+\cos\tfrac{\pi}{6})})}=2\sqrt{2(1+2\cos\tfrac{\pi}{12})}=4\cos\tfrac{\pi}{24}.$$So take $a=4,\,b=24$.

Answer 2

Hint: Your term is equal to $$2\sqrt{2+2\sqrt{2+\sqrt{3}}}$$