Explanation

(c)
Given equation can be written as
$\left( 1 + \cos\frac{\pi}{8} \right)\left( 1 + \cos\frac{7\pi}{8} \right)\left( 1 + \cos\frac{3\pi}{8} \right)\left( 1 + \cos\frac{5\pi}{8} \right)$
$= \left( 1 + \cos{\frac{\pi}{8} + \cos{\frac{7\pi}{8} + \cos{\frac{\pi}{8}\cos\frac{7\pi}{8}}}} \right)\left( 1 + \cos{\frac{3\pi}{8} + \cos{\frac{5\pi}{8} + \cos\frac{3\pi}{8}\cos\frac{5\pi}{8}}} \right)$
$= \left( 1 + \cos{\frac{\pi}{8} - \cos{\frac{\pi}{8} + \cos{\frac{\pi}{8}\cos\frac{7\pi}{8}}}} \right)\left( 1 - \cos{\frac{5\pi}{8} + \cos{\frac{5\pi}{8} + \cos\frac{3\pi}{8}\cos\frac{5\pi}{8}}} \right)$
$= \left( 1 + \cos{\frac{\pi}{8}\cos\frac{7\pi}{8}} \right)\left( 1 + \cos{\frac{3\pi}{8}\cos\frac{5\pi}{8}} \right)$
$= \frac{1}{4}\left( 2 + 2\cos{\frac{\pi}{8}\cos\frac{7\pi}{8}} \right)\left( 2 + 2\cos{\frac{3\pi}{8}\cos\frac{5\pi}{8}} \right)$
$= \frac{1}{4}\left( 2 + \cos{\frac{3\pi}{4} + \cos\pi} \right)\left( 2 + \cos{\frac{\pi}{4}\cos\pi} \right)$
$= \frac{1}{4}\left( 1 + \cos\frac{3\pi}{4} \right)\left( 1 + \cos\frac{\pi}{4} \right)$
$= \frac{1}{4}\left( 1 - \cos\frac{\pi}{4} \right)\left( 1 + \cos\frac{\pi}{4} \right)$
$= \frac{1}{4}\left( 1 - \cos^{2}\frac{\pi}{4} \right) = \frac{1}{4}\left( 1 - \frac{1}{2} \right) = \frac{1}{8}$