# How do you express sin(pi/ 8 ) * cos(( pi / 4 )  without using products of trigonometric functions?

Dec 31, 2016

$\left(\frac{\sqrt{2}}{2}\right) \left(\frac{\sqrt{2 - \sqrt{2}}}{2}\right)$

#### Explanation:

$P = \sin \left(\frac{\pi}{8}\right) . \cos \left(\frac{\pi}{4}\right)$
Trig table gives $\cos \left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}$, then P can be expressed as:
$P = \left(\frac{\sqrt{2}}{2}\right) \sin \left(\frac{\pi}{8}\right)$.
We can evaluate $\sin \left(\frac{\pi}{8}\right)$ by applying the trig identity:
$2 {\sin}^{2} a - 1 - \cos 2 a$
$2 {\sin}^{2} \left(\frac{\pi}{8}\right) = 1 - \cos \left(\frac{\pi}{4}\right) = 1 - \frac{\sqrt{2}}{2} = \frac{2 - \sqrt{2}}{2}$
${\sin}^{2} \left(\frac{\pi}{8}\right) = \frac{2 - \sqrt{2}}{4}$
$\sin \left(\frac{\pi}{8}\right) = \pm \left(\frac{\sqrt{2 - \sqrt{2}}}{2}\right)$
Since $\sin \left(\frac{\pi}{8}\right)$ is positive, take the positive value.
Finally:
$P = \left(\frac{\sqrt{2}}{2}\right) \left(\frac{\sqrt{2 - \sqrt{2}}}{2}\right)$