Toss the coin 2017 times

There is a biased coin whose probability of getting a head at the \(i^\text{th}\) flip in a game of 2017 flips be given by \(\sin^2\left(\frac{i\pi}{2017}\right)\).

If the variance of this probability distribution can be expressed as \( \frac pq\), where \(p\) and \(q\) are coprime positive integers, find \(p+q\).

\(\)
Bonus: Generalize this.

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