Example of Parseval's Theorem In the textbook "Mathematics for Physics" of Stone and Goldbart the following example for an illustration of Parseval's Theorem is given:


Until 2.42 I understand everything but I don't understand the statement:
" Finally, as $\sin^2(\pi(\zeta-n))=\sin^2(\pi \zeta)$ "
Can you explain me why this equality holds?

$\sin{\pi n} = 0$ when $n \in \mathbb{Z}$.  Thus, because $\cos{\pi n} = (-1)^n$, we have
$$\sin{\pi (\zeta - n)} = \sin{\pi \zeta} \cos{\pi n} - \sin{\pi n} \cos{\pi \zeta} = (-1)^n \sin{\pi \zeta} $$
