# How do you express cos( (15 pi)/ 8 ) * cos (( 5 pi) /8 )  without using products of trigonometric functions?

Apr 4, 2016

$\cos \left(\frac{15 \pi}{8}\right) \cos \left(\frac{5 \pi}{8}\right) = \frac{1}{2} \cos \left(\frac{5 \pi}{2}\right) + \frac{1}{2} \cos \left(\frac{5 \pi}{4}\right) = - \frac{\sqrt{2}}{2}$

#### Explanation:

$2 \cos A \cos B = \cos \left(A + B\right) + \cos \left(A - B\right)$

$\cos A \cos B = \frac{1}{2} \left(\cos \left(A + B\right) + \cos \left(A - B\right)\right)$

$A = \frac{15 \pi}{8} , B = \frac{5 \pi}{8}$

$\implies \cos \left(\frac{15 \pi}{8}\right) \cos \left(\frac{5 \pi}{8}\right) = \frac{1}{2} \left(\cos \left(\frac{15 \pi}{8} + \frac{5 \pi}{8}\right) + \cos \left(\frac{15 \pi}{8} - \frac{5 \pi}{8}\right)\right)$

$= \frac{1}{2} \left(\cos \left(\frac{20 \pi}{8}\right) + \cos \left(\frac{10 \pi}{8}\right)\right)$

$= \frac{1}{2} \cos \left(\frac{5 \pi}{2}\right) + \frac{1}{2} \cos \left(\frac{5 \pi}{4}\right) = 0 + - \frac{\sqrt{2}}{2} = - \frac{\sqrt{2}}{2}$

$\cos \left(\frac{15 \pi}{8}\right) \cos \left(\frac{5 \pi}{8}\right) = \frac{1}{2} \cos \left(\frac{5 \pi}{2}\right) + \frac{1}{2} \cos \left(\frac{5 \pi}{4}\right) = - \frac{\sqrt{2}}{2}$