Is $(\sin{x})(\sin{\pi x})$ periodic?

Using the product-to-sum formula, $$ \sin x \sin\pi x = \frac{1}{2} \left( \cos ((1-\pi)x) - \cos((1 + \pi)x) \right) $$ but $\frac{1-\pi}{1+\pi} \not\in \mathbb{Q}$ and $\sin x \sin \pi$ is continuous, so this function is not periodic.

If $f(x)=\sin(x)\sin(\pi x)$ were periodic, then in particular its set of zeroes would be periodic. But the zero set is $\mathbb Z\cup\pi\mathbb Z$, and $a\mathbb Z\cup b\mathbb Z$ (for nonzero $a$, $b$) is not periodic unless $b/a$ is rational.

(Suppose $a\mathbb Z\cup b\mathbb Z$ is periodic with period $P>0$. Then $0+P\in a\mathbb Z\cup b\mathbb Z$; assume without loss of generality that $P=ka$ for some $k\in Z$. Now $b+P = ka+b$ must be in $a\mathbb Z\cup b\mathbb Z$ too, but if $ka+b=bn$ then $\frac ba=\frac{k}{n-1}$, and if $ka+b=am$, then $\frac ba = m-k$; in both cases $\frac ba$ is rational.)