I'm trying to find out why I can get $$\sin(x)=x\cdot\lim_{n\to\infty}\prod_{k=1}^{\frac{n-1}{2}}\left[1-\frac{\big(1+\cos(\frac{2k\pi}{n})\big)\cdot{x^2}}{\big(1-\cos(\frac{2k\pi}{n})\big)\cdot{n^2}}\right]$$ through factorization of $$\sin(x)=x\cdot\lim_{n\to\infty}\sum_{k=0}^{\frac{n-1}{2}}(-1)^k{{2k+1}\choose{n}}\frac{x^{2k}}{n^{2k+1}}$$.
How is the product derived from the sum?