Trigonometric Series : Trinity

Calculus Level 5

\[\displaystyle 1 + \dfrac{2}{3}\sin^2x + \dfrac{2 \cdot 4}{3 \cdot 5}\sin^4x + \dfrac{2 \cdot 4 \cdot 6}{3 \cdot 5 \cdot 7}\sin^6x + \ldots\]

Let \(f(x) \) denote the value of the above expression.

Then \(\displaystyle f\left( \dfrac{\pi}{24} \right)\) can be written as

\[\dfrac{\pi^a (\sqrt{b} + c)}{d\sqrt{e}}\]

where \(b,e\) are squarefree and \(a,b,c,d,e \in \mathbb N\). Find the value of \(a+b+c+d+e\).

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