Unknown identities used in Fourier series

Question

This step was found in a solution manual for an exercise. How is it achieved? $\sum_{n=1}^{\infty} \frac{1-(-1)^n}{n^2} cos(nt) = \sum_{n=1}^{\infty} \frac{2}{(2n-1)^2} cos[(n-1)t]$

Answer 1

Hint: Consider $n$ as odd $n=2k-1$ and even $n=2k$ indices separately, you find $$\sum_{n=1}^{\infty} \frac{1-(-1)^n}{n^2} \cos nt = \sum_{k=1}^{\infty} \frac{2}{(2k-1)^2} \cos(2k-1)t$$