# Question 3394e

Apr 7, 2017

$L H S = \sin \left(\frac{3 \pi}{14}\right) - \sin \left(\frac{\pi}{14}\right) - \sin \left(\frac{5 \pi}{14}\right)$

$= \sin \left(\frac{3 \pi}{14}\right) - \left(\sin \left(\frac{\pi}{14}\right) + \sin \left(\frac{5 \pi}{14}\right)\right)$

$= \sin \left(\frac{3 \pi}{14}\right) - \left(2 \sin \left(\frac{3 \pi}{14}\right) \cos \left(\frac{2 \pi}{14}\right)\right)$

$= \sin \left(\frac{3 \pi}{14}\right) - \left(2 \cos \left(\frac{\pi}{2} - \frac{3 \pi}{14}\right) \cos \left(\frac{2 \pi}{14}\right)\right)$

$= \sin \left(\frac{3 \pi}{14}\right) - \left(2 \cos \left(\frac{7 \pi - 3 \pi}{14}\right) \cos \left(\frac{2 \pi}{14}\right)\right)$

$= \sin \left(\frac{3 \pi}{14}\right) - \left(2 \cos \left(\frac{4 \pi}{14}\right) \cos \left(\frac{2 \pi}{14}\right)\right)$

$= \sin \left(\frac{3 \pi}{14}\right) - \frac{1}{2 \sin \left(\frac{2 \pi}{14}\right)} \left(2 \cos \left(\frac{4 \pi}{14}\right) 2 \sin \left(\frac{2 \pi}{14}\right) \cos \left(\frac{2 \pi}{14}\right)\right)$

$= \sin \left(\frac{3 \pi}{14}\right) - \frac{1}{2 \sin \left(\frac{2 \pi}{14}\right)} \left(2 \cos \left(\frac{4 \pi}{14}\right) \sin \left(\frac{4 \pi}{14}\right)\right)$

$= \sin \left(\frac{3 \pi}{14}\right) - \frac{1}{2 \sin \left(\frac{2 \pi}{14}\right)} \left(\sin \left(\frac{8 \pi}{14}\right)\right)$

$= \sin \left(\frac{3 \pi}{14}\right) - \frac{1}{2 \sin \left(\frac{2 \pi}{14}\right)} \left(\sin \left(\pi - \frac{6 \pi}{14}\right)\right)$

$= \sin \left(\frac{3 \pi}{14}\right) - \frac{1}{2 \sin \left(\frac{2 \pi}{14}\right)} \left(\sin \left(\frac{6 \pi}{14}\right)\right)$

$= \sin \left(\frac{3 \pi}{14}\right) - \frac{1}{2 \sin \left(\frac{2 \pi}{14}\right)} \left(\sin \left(\frac{3 \cdot 2 \pi}{14}\right)\right)$

$= \sin \left(\frac{3 \pi}{14}\right) - \frac{1}{2 \sin \left(\frac{2 \pi}{14}\right)} \left(3 \sin \left(\frac{2 \pi}{14}\right) - 4 {\sin}^{3} \left(\frac{2 \pi}{14}\right)\right)$

$= \sin \left(\frac{3 \pi}{14}\right) - \frac{1}{2} \left(3 - 2 \cdot 2 {\sin}^{2} \left(\frac{2 \pi}{14}\right)\right)$

=sin ((3pi)/14)-1/2(3-2(1-cos((4pi)/14)) #

$= \sin \left(\frac{3 \pi}{14}\right) - \frac{1}{2} \left(3 - 2 + 2 \cos \left(\frac{4 \pi}{14}\right)\right)$

$= \sin \left(\frac{3 \pi}{14}\right) - \frac{1}{2} \left(1 + 2 \cos \left(\frac{4 \pi}{14}\right)\right)$

$= \sin \left(\frac{3 \pi}{14}\right) - \frac{1}{2} - \cos \left(\frac{4 \pi}{14}\right)$

$= \sin \left(\frac{3 \pi}{14}\right) - \frac{1}{2} - \sin \left(\frac{\pi}{2} - \frac{4 \pi}{14}\right)$

$= \sin \left(\frac{3 \pi}{14}\right) - \frac{1}{2} - \sin \left(\frac{7 \pi - 4 \pi}{14}\right)$

$= \cancel{\sin \left(\frac{3 \pi}{14}\right)} - \frac{1}{2} - \cancel{\sin \left(\frac{3 \pi}{14}\right)}$

$= - \frac{1}{2}$

Proved