We're Only On Coprime Numbers?

Geometry Level 5

Evaluate

\[\large P=\prod_{k}\sin\left(\dfrac{k\pi}{2016}\right)\]

where the product is taken over all \(k\) with \(1\leq k\leq 1008\) and \(\gcd(k,2016)=1\).

Write your answer in the form \(P=\dfrac{a}{2^b}\), where \(a\) and \(b\) are integers, and \(a\) is odd. Enter \(a+b\).

Clariifcation: \( \gcd(\cdot) \) denotes the greatest common divisor function.


Inspiration.

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