Inscribed To Infinity

The equation relating the inradius and the circumradius of a regular polygon, $r=R\cos\bigg(\frac{\pi}{n}\bigg)$ gives the ratio of the radii of the final to initial circles as $\large K^{'}=\frac{r_\text{final circle}}{r_\text{initial circle}}=\cos\bigg(\frac{\pi}{3}\bigg)\cos\bigg(\frac{\pi}{4}\bigg)\cos\bigg(\frac{\pi}{5}\bigg)\cdots$ Numerically, $K^{'}=\frac{1}{K}=\frac{1}{8.7000366252}=0.1149420448$

Kepler-Bouwkamp constant.

Got on Wikipedia

Can anyone see a nice way to solve this analytically?

I reluctantly gave up and used Python to approximate the product series

$\prod_{n=3}^{n=10000000}\sin(\frac{(n-2)\pi}{2n}) = 0.11494210157473356$

Here is the code :

import numpy as np
def product_series(N):
product = 1.0
for i in range(3,n+1):
    product*= np.sin((n-2)*np.pi/(2*n))
return product

product_series(10000000)