complex analysis - Prove that $sin(pi z)$ can be written as infinite product

Prove that
\begin{align}
\sin(\pi z) = \pi z \prod_{n=1}^{\infty} \left( 1-\frac{z^2}{n^2}\right) \, \, \, \, \forall \, z \in \mathbb{C}
\end{align}

The hint I had it's to use the Fourier series, but I really don't see how.
Any suggestions? Thanks in advance!