Trigonometry: Half-Angle Identities

If $a$ and $b$ are positive integers such that $$\sqrt{8 + \sqrt{32 + \sqrt{768}}} = a \cos \frac{\pi}{b},$$ compute the ordered pair $(a, b)$. Write your answer as (a,b), without spaces.

(Hint: Can you express $\sqrt{768}$ nicely in terms of cosine?)

Source: Math Prize for Girls