Originally Posted by

**bkarpuz** Dear Friends,

I have a question about the sine product formulae.

As far as I know, Euler thinks the function $\displaystyle \sin$ as a polynomial of $\displaystyle \infty$ order, and writes it by using its roots as

$\displaystyle \sin(x)=a_{0}\prod_{k=-\infty}^{\infty}\big(x-k\pi\big)$

or equivalently

$\displaystyle \sin(x)=b_{0}x\prod_{k=1}^{\infty}\bigg(1-\Big(\frac{x}{k\pi}\Big)^{2}\bigg)$........(1)

for some constants $\displaystyle a_{0},b_{0}$.

From the fact that

$\displaystyle \lim_{x\to0}\frac{\sin(x)}{x}=1,$

the constant $\displaystyle b_{0}$ in (1) is computed to be $\displaystyle 1$.

Hence

$\displaystyle \sin(x)=x\prod_{k=1}^{\infty}\bigg(1-\Big(\frac{x}{k\pi}\Big)^{2}\bigg)$........(2)

My question comes at this point.

How I can be sure that the right-hand side of (2) gives $\displaystyle \sin(x)$, not $\displaystyle \big(\sin(x)\big)^{2}/x$?