Trigonometric Series : Trinity

Calculus Level 5

1+23sin2x+2435sin4x+246357sin6x+\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)f(x) denote the value of the above expression.

Then f(π24)\displaystyle f\left( \dfrac{\pi}{24} \right) can be written as

πa(b+c)de\dfrac{\pi^a (\sqrt{b} + c)}{d\sqrt{e}}

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

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