Fault in proof of $\zeta(2) = \frac{\pi^2}{6}$

Question

Consider the proof of: $$\zeta(2) = \frac{\pi^{2}}{6}$$

So the proof assume that (because of Euler decomposition) $$\frac{\sin(x)}{x} = \prod_{n > 0}\left(1 - \frac{x^{2}}{(n\pi)^{2}}\right)$$

Why should we consider the $\frac{\sin(x)}{x}$, that $\sin(x)$ has the same decomposition?

Please tell me where's my fault.