What is the value of $\sin^2 (\frac{\pi}{10}) \sin^2 (\frac{3\pi}{10})$? 
PROBLEM
  $$ \prod_{i=0}^4 \left(1 + \cos \left(\frac{(2k+1)\pi}{10}\right)\right)$$

My Try
$$
\left(1 + \cos \frac{\pi}{10}\right)
\left(1 + \cos \frac{9\pi}{10}\right)
\left(1 + \cos \frac{7\pi}{10}\right)
\left(1 + \cos \frac{3\pi}{10}\right)
= \sin^2 \left(\frac{\pi}{10}\right)
  \sin^2 \left(\frac{3\pi}{10}\right)
$$
I am not able to proceed further. 
Please help me.

$$\sin^2 {\frac {\pi}{10}} \sin^2 \frac {3\pi}{10}=\left(\frac {4\sin {\frac {\pi}{5}}. \cos {\frac {2\pi}{10}}\cos {\frac {4\pi}{10}}}{4\sin \frac {\pi}{5}}\right)^2=\left(\frac {\sin \frac {4\pi}{5}}{4\sin \frac {\pi}{5}}\right)^2=\frac {1}{16}$$
