Product $\prod _{n=1}^{\infty } \frac{1}{2} \sqrt{\frac{4 (\pi n-x) (\pi n+x)}{\pi ^2 n^2-4 x^2}}$

Question

$$\prod _{n=1}^{\infty } \frac{1}{2} \sqrt{\frac{4 (\pi n-x) (\pi n+x)}{\pi ^2 n^2-4 x^2}}=\frac{\sqrt{2}}{\sqrt[4]{1+e^{-2 i x}} \sqrt[4]{1+e^{2 i x}}}$$ Mathematica Could not find the solution it is possible to show?.

Answer 1

Hint. One may recall the Weierstrass product of the sine function $$ \frac{\sin x}{x}=\prod\limits_{n=1}^\infty\left(1-\frac{x^2}{\pi^2 n^2}\right), \qquad x\in(0,\pi), $$ then one may write, for appropriate values of $x$, $$ \prod _{n=1}^{\infty } \frac{1}{2} \sqrt{\frac{4 (\pi n-x) (\pi n+x)}{\pi ^2 n^2-4 x^2}}=\sqrt{\frac{\prod _{n=1}^{\infty }\left(1-\frac{x^2}{\pi^2 n^2}\right)}{\prod _{n=1}^{\infty }\left(1-\frac{4x^2}{\pi^2 n^2}\right)}}=\sqrt{\frac{\frac{\sin x}{x}}{\frac{\sin (2x)}{2x}}}=\frac1{\sqrt{\cos x}}. $$