I recently started learning about Fourier series, so I'm still kind of shaky on the topic
Given $f(t)=f(t+T)$ and $$f(t)=\sum_{n=-\infty}^{\infty}F_ne^{jn \omega_0t}$$ Show that:
(a) If $f(t)=f(t+\frac{T}{2})$, then $F_n=0$ for $n$ odd.
(b) If $f(t)=-f(t+\frac{T}{2})$, then $F_n=0$ for $n$ even.
I'm not sure how to go about proving this analytically; I know that $f(t)$ is periodic with period T, so for an $f(t) = f(t + \frac{T}{2})$ would have the function in the middle of its period.
Any help would be appreciated! Thanks.