A nice Trignometric Series

Calculus Level 4

Let

f(x)=sin2x12sin2x×sin2x+13sin3x×sin3x f(x) = \sin^{2} x - \frac{1}{2} \sin2x \times \sin^{2}x + \frac{1}{3} \sin3x \times \sin^{3}x - \ldots

Then f(π12) \displaystyle f \left( \frac{\pi}{12} \right) can be written as

tan1(abc) \tan^{-1} \left( \dfrac{ a - \sqrt{b}}{c} \right)

where bb is square free & a,b,cNa,b,c \in\mathbb N. Find the value of a+b+ca+b+c.

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