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? 

$$\left( 1+\cos { \left( \frac { \pi  }{ 8 }  \right)  }  \right) \left( 1+\cos { \left( \frac { 3\pi  }{ 8 }  \right)  }  \right) \left( 1+\cos { \left( \frac { 5\pi  }{ 8 }  \right)  }  \right) \left( 1+\cos { \left( \frac { 7\pi  }{ 8 }  \right)  }  \right) =\\ =\left( 1+\cos { \left( \frac { \pi  }{ 8 }  \right)  }  \right) \left( 1+\cos { \left( \frac { 3\pi  }{ 8 }  \right)  }  \right) \left( 1+\cos { \left( \pi -\frac { 3\pi  }{ 8 }  \right)  }  \right) \left( 1+\cos { \left( \pi -\frac { \pi  }{ 8 }  \right)  }  \right) =\\ =\left( 1+\cos { \left( \frac { \pi  }{ 8 }  \right)  }  \right) \left( 1+\cos { \left( \frac { 3\pi  }{ 8 }  \right)  }  \right) \left( 1-\cos { \left( \frac { 3\pi  }{ 8 }  \right)  }  \right) \left( 1-\cos { \left( \frac { \pi  }{ 8 }  \right)  }  \right) =\\ =\left( 1-\cos ^{ 2 }{ \frac { \pi  }{ 8 }  }  \right) \left( 1-\cos ^{ 2 }{ \frac { 3\pi  }{ 8 }  }  \right) =\sin ^{ 2 }{ \frac { \pi  }{ 8 }  } \sin ^{ 2 }{ \frac { 3\pi  }{ 8 }  } =\frac { \left( 1-\cos { \frac { \pi  }{ 4 }  }  \right)  }{ 2 } \cdot \frac { \left( 1-\cos { \frac { 3\pi  }{ 4 }  }  \right)  }{ 2 } =\\ =\frac { \left( 1-\cos { \frac { \pi  }{ 4 }  }  \right) \left( 1-\cos { \left( \pi -\frac { \pi  }{ 4 }  \right)  }  \right)  }{ 4 } =\frac { \left( 1-\cos { \frac { \pi  }{ 4 }  }  \right) \left( 1+\cos { \frac { \pi  }{ 4 }  }  \right)  }{ 4 } =\frac { 1-\cos ^{ 2 }{ \frac { \pi  }{ 4 }  }  }{ 4 } =\frac { 1-\frac { 1 }{ 2 }  }{ 4 } =\frac { 1 }{ 8 }  $$
