# How do you express sin(pi/12) * cos(( 7 pi)/12 )  without products of trigonometric functions?

Jul 1, 2018

$- \frac{1}{2} + \frac{\sqrt{3}}{4}$

#### Explanation:

$\sin \left(\frac{\pi}{12}\right) = \frac{\sqrt{3} - 1}{2 \sqrt{2}}$
$\cos \left(\frac{7 \pi}{12}\right) = - \frac{\sqrt{3} - 1}{2 \sqrt{2}}$

$\sin \left(\frac{\pi}{12}\right) \cdot \cos \left(\frac{7 \pi}{12}\right) = - {\left(\frac{\sqrt{3} - 1}{2 \sqrt{2}}\right)}^{2}$

$- {\left(\frac{\sqrt{3} - 1}{2 \sqrt{2}}\right)}^{2} = - \frac{1}{8} {\left(\sqrt{3} - 1\right)}^{2} = - \frac{3}{8} + \frac{\sqrt{3}}{4} - \frac{1}{8} = - \frac{1}{2} + \frac{\sqrt{3}}{4}$

Jul 1, 2018

color(indigo)(=> (sqrt3 - 2) / 4

#### Explanation:

$\sin \left(\frac{\pi}{12}\right) \cdot \cos \left(\frac{7 \pi}{12}\right)$

=> (1/2) (sin (pi/12 + (7pi)/12) + (sin (pi/12) - ((7pi)/12))

=> (1/2) (sin ((2pi)/3) + sin (-(pi/2))

$\implies \left(\frac{1}{2}\right) \left(\sin \left(\frac{\pi}{3}\right) - \sin \left(\frac{\pi}{2}\right)\right)$

$\implies \left(\frac{1}{2}\right) \left(\frac{\sqrt{3}}{2} - 1\right)$

color(indigo)(=> (sqrt3 - 2) / 4