$\zeta(2)$ Euler's proof (Basel problem)

Question

At one point Euler assumes that $$\frac{\sin x}{x} = \prod_{n=1}^{\infty}\left(1-\frac{x}{n\pi} \right)\left(1-\frac{-x}{n\pi} \right)$$ Why does he assume that? If we factor random functions according to their roots, we can conclude $$e^x \frac{\sin x}{x}= \prod_{n=1}^{\infty}\left(1-\frac{x}{n\pi} \right)\left(1-\frac{-x}{n\pi} \right)=\frac{\sin x}x$$