The value of (1 + cos(π)/(8))(1 + cos (3π)/(8))(1 + cos(5π)/(8))(1...

Given expression can be written as

\left(1 + cos \dfrac{\pi}{8}\right) \left(1 + cos \dfrac{7\pi}{8}\right)\left(1 + cos \dfrac{3\pi}{8}\right) \left(1 + cos \dfrac{5\pi}{8}\right)<br /><br />= \left(1 + \text{cos}\dfrac{π}{8}\right)\left(1 - \text{cos}\dfrac{π}{8}\right) \left(1 + cos\dfrac{3\pi}{8}\right)\left($1 -$\text{cos}\dfrac{3\pi}{8}\right)

$= \left(1 - \cos^{2}\dfrac{π}{8}\right)\left(1 - \cos^{2}\dfrac{3\pi}{8}\right)$

$= \left(\sin^{2}\dfrac{π}{8}\right) \left(\sin^{2}\dfrac{3\pi}{8}\right)<br /><br />$= \sin^{2} $\dfrac{\pi}{8} \cdot \cos^{2} \dfrac{\pi}{8}<br /><br />=\dfrac{1}{4} {\left(2 \text{sin} \dfrac{\pi}{8} \dfrac{\text{cos} \pi}{8}\right)}^{2}<br /><br />=\dfrac{1}{4} \sin^{2} \dfrac{\pi}{4}<br /><br />=\dfrac{1}{8}$