Let's do some calculus! (37)

Calculus Level 5

$\sum_{n=0}^{\infty} \dfrac{ \cos \bigg( \dfrac{3 \pi}{5} \left(2n+1 \right) \bigg) + \cos \bigg( \dfrac{\pi}{5} \left(2n+1 \right) \bigg)}{{\left(2n+1 \right)}^2}$

The infinite sum above can be represented as $$\dfrac{{\pi}^a}{b}$$, where $$a$$ and $$b$$ are non co-prime positive integers. Find the value of $$\dfrac ba$$.

This problem is inspired from Ishan Singh.