Radically Compromised

Geometry Level 5

$\large a_{n+1} = \sqrt{2+a_n}$

Consider the recurrence relation $a_n$ above for $n = 1,2, 3, \ldots$ and $a_1 = \sqrt3$ .

Evaluate $\log_2 \left [ \cos^{-1} \left ( \dfrac{a_{2016}}2 \right) \right ]$ .

If this number can be expressed as $\log_2 \pi - \log_2 a - b$ , where $a$ and $b$ are positive integers with $a$ odd, submit your answer as $a+b$ .