I am looking to show that:
$$\sinh(x)/x=\lim_{n\rightarrow \infty} \frac{1}{n}\prod^{n-1}_{k=1} \left( \frac{x^2}{n^2}-2(\cos\left(\frac{k \pi}{n}\right)-1)\right)\tag{1}$$ which I know to be true from other means.
I have tried using that: $$ \sinh(x)/x= \prod^\infty_{n=1}\left(1+\frac{x^2}{k^2 \pi^2}\right)$$ but naively expanding the $\cos(\ldots)$ in (1) doesn't work. Does anyone know the best way to approach this?