I think you are talking about the Gamma reflection formula

$\displaystyle \Gamma(z)\Gamma(1-z)=\frac{\pi}{\sin(\pi z)}.$

Actually, this is not how SPF (sine product formula) is obtained, it is rather in this way if you trace the proof

$\displaystyle \sin(\pi z)=\frac{\pi}{\Gamma(z)\Gamma(1-z)},$

and as you have mentioned the proof makes use of Weierstrass' Gamma function expansion

$\displaystyle \Gamma(z):=\frac{\mathrm{e}^{-\gamma z}}{z}\prod_{k\in\mathbb{N}}\frac{\mathrm{e}^{z/k}}{1+(z/k)},$

where $\displaystyle \gamma$ is the Euler–Mascheroni constant defined by

$\displaystyle \gamma:=\lim_{n\to\infty}\Big(\sum_{k=1}^{n}\frac{ 1}{k}-\int_{1}^{n}\frac{\mathrm{d}x}{x}\Big).$

This is where my interest to the SPF originates.