Let $f$ and $g$ be periodic functions of period $p$. Then $af(x)+bf(x)$ with $a,b$ constants and $f(x)g(x)$ are both of period $p$
I'm not exactly sure how to prove these properties of periodic functions. I think I may have proven the first one, but as I remarked, I'm not sure.
To prove the first one, $f(x)=f(x+p)$. So $a f(x+p)+bg(x+p)=af(x)+bg(x)$.
The same approach works for the other example: $f(x+p)g(x+p)=f(x)g(x+p)=f(x)g(x)$. The first equality follows from the $p$-periodicity of $f$, the second from that of $g$.