Infinite product of trigonometric function

Question

I would like to find the infinite product of $\frac {\sin x\pi}{\sin \sqrt{x}\pi}$. I have tried to seperate them into two parts namely $$\prod_{n=1}^{\infty}\frac{n^2-x^2}{n^2-x}$$ but it seems to not fit the o.g. question. How can I do this?
Tips are accepted.

Answer 1

You have $$\frac{\sin (\pi x)}{\pi x}=\prod_{n=1}^\infty \left(1-\frac{x^2}{n^2}\right)\qquad \text{and} \qquad \frac{\sin \left(\pi \sqrt{x}\right)}{\pi \sqrt{x}}=\prod_{n=1}^\infty \left(1-\frac{x}{n^2}\right)$$ Making the ratios $$\frac{\sin (\pi x)}{\sin \left(\pi \sqrt{x}\right)}=\sqrt{x}\prod_{n=1}^\infty \frac{n^2-x^2}{n^2-x}$$