A nice Trignometric Series

Calculus Level 4

Let

\[ f(x) = \sin^{2} x - \frac{1}{2} \sin2x \times \sin^{2}x + \frac{1}{3} \sin3x \times \sin^{3}x - \ldots\]

Then \( \displaystyle f \left( \frac{\pi}{12} \right) \) can be written as

\[ \tan^{-1} \left( \dfrac{ a - \sqrt{b}}{c} \right)\]

where \(b\) is square free & \(a,b,c \in\mathbb N\). Find the value of \(a+b+c\).

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