Evaluating $(1+\cos\frac{\pi}{8})(1+\cos\frac{3\pi}{8}) (1+\cos\frac{5\pi}{8})(1+\cos\frac{7\pi}{8})$ 
How to find the value of $$\left(1+\cos\frac{\pi}{8}\right)\left(1+\cos\frac{3\pi}{8}\right) \left(1+\cos\frac{5\pi}{8}\right)\left(1+\cos\frac{7\pi}{8}\right)$$


I used: $ 1+ \cos \theta=2\sin^2(\theta/2)$ to get $$2^4 \cdot\prod_{n=0}^3 \sin\left(\frac{(2n+1)\pi}{16}\right)$$
After that I tried using the formula: 
$$\sin\left(1 \frac{\pi}n\right)\sin\left(2 \frac{\pi}n\right) \cdots \sin\left((n-1) \frac{\pi}n\right)= \frac n {2^{n-1}}$$
to find the value of the expression in the parenthesis in the first equation, but currently I'm stuck. I don't think this method will lead me to the correct answer.
Any help would be appreciated!

Use the linearisation formulae. The expression can be rewritten as:
 $$2^2\Bigl(2\sin\frac{\pi}{16}\,\sin\frac{7\pi}{16}\Bigr)\Bigl(2\sin\frac{3\pi}{16}\,\sin\frac{5\pi}{16}\Bigr)=2^2\cos\frac{3\pi}{8}\cos\frac{\pi}{8}=2\cos\frac{\pi}{4}=\sqrt2.$$
