# How do you express cos( (5 pi)/6 ) * cos (( 5 pi) /12 )  without using products of trigonometric functions?

Mar 18, 2016

$P = - \left(\frac{\sqrt{3}}{4}\right) \left(\sqrt{2 - \sqrt{3}}\right)$

#### Explanation:

$P = \cos \left(\frac{5 \pi}{6}\right) . \cos \left(\frac{5 \pi}{12}\right)$
Trig table --> $\cos \left(\frac{5 \pi}{6}\right) = - \frac{\sqrt{3}}{2}$
Find $\cos \left(\frac{5 \pi}{12}\right)$ by the identity: $\cos 2 a = 2 {\cos}^{2} a - 1$
$\cos \left(\frac{5 \pi}{6}\right) = - \frac{\sqrt{3}}{2} = 2 {\cos}^{2} \left(\frac{5 \pi}{12}\right) - 1$
$2 {\cos}^{2} \left(\frac{5 \pi}{12}\right) = 1 - \frac{\sqrt{3}}{2} = \frac{2 - \sqrt{3}}{2}$
${\cos}^{2} \left(\frac{5 \pi}{12}\right) = \frac{2 - \sqrt{3}}{4}$
$\cos \left(\frac{5 \pi}{12}\right) = \frac{\sqrt{2 - \sqrt{3}}}{2}$ -->$\cos \left(\frac{5 \pi}{12}\right)$ is positive.
Finally,
$P = \left(- \frac{\sqrt{3}}{2}\right) \left(\frac{\sqrt{2 - \sqrt{3}}}{2}\right) = - \left(\frac{\sqrt{3}}{4}\right) \left(\sqrt{2 - \sqrt{3}}\right)$