What is the value of the expression $[1 + \cos(\frac{\pi}{8})][1 + \cos(\frac{3\pi}{8})][1 + \cos(\frac{5\pi}{8})][1 + \cos(\frac{7\pi}{8})]$? This is rather a simple problem that I'm posting ; looking forward not for the solution of it but the different ways it could be solved.
What is the value of the following expression? 

$$\left( 1+\cos { \left( \frac { \pi  }{ 8 }  \right)  }  \right) \left( 1+\cos { \left( \frac { 3\pi  }{ 8 }  \right)  }  \right) \left( 1+\cos { \left( \frac { 5\pi  }{ 8 }  \right)  }  \right) \left( 1+\cos { \left( \frac { 7\pi  }{ 8 }  \right)  }  \right) $$

Edit: Do I manually put the values of $\cos\frac{\pi}{8}$ and all other cosine terms? Is there a better and shorter way? 

Like The value of $\cos^4\frac{\pi}{8} + \cos^4\frac{3\pi}{8}+\cos^4\frac{5\pi}{8}+\cos^4\frac{7\pi}{8}$,
if $y=1+c,$ we have $$8(y-1)^4-8(y-1)^2+1=0\iff8y^4+\cdots+1=0$$
$$\implies\prod_{n=0}^3\left(1+\cos\dfrac{(2n+1)\pi}8\right)=\dfrac18$$
