# How do you prove that cot^2(pi/7)+cot^2((2pi)/7)+cot^2((3pi)/7)=5?

Mar 16, 2018

${\cot}^{2} \left(\frac{\pi}{7}\right) + {\cot}^{2} \left(\frac{2 \pi}{7}\right) + {\cot}^{2} \left(\frac{3 \pi}{7}\right)$

$= {\csc}^{2} \left(\frac{\pi}{7}\right) + {\csc}^{2} \left(\frac{2 \pi}{7}\right) + {\csc}^{2} \left(\frac{3 \pi}{7}\right) - 3$

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

$= \frac{2}{1 - \cos \left(\frac{2 \pi}{7}\right)} + \frac{2}{1 - \cos \left(\frac{4 \pi}{7}\right)} + \frac{2}{1 - \cos \left(\frac{6 \pi}{7}\right)} - 3$

$= \frac{2}{1 + \cos \left(\frac{5 \pi}{7}\right)} + \frac{2}{1 + \cos \left(\frac{3 \pi}{7}\right)} + \frac{2}{1 + \cos \left(\frac{\pi}{7}\right)} - 3$

$= 2 \left[\frac{\left(1 + \cos \left(\frac{3 \pi}{7}\right)\right) \left(1 + \cos \left(\frac{\pi}{7}\right)\right) + \left(1 + \cos \left(\frac{5 \pi}{7}\right)\right) \left(1 + \cos \left(\frac{\pi}{7}\right) + \left(1 + \cos \left(\frac{5 \pi}{7}\right)\right) \left(1 + \cos \left(\frac{3 \pi}{7}\right)\right)\right)}{\left(1 + \cos \left(\frac{5 \pi}{7}\right)\right) \left(1 + \cos \left(\frac{3 \pi}{7}\right)\right) \left(1 + \cos \left(\frac{\pi}{7}\right)\right)}\right] - 3$

$= 2 \left[\frac{3 + 2 \cos \left(\frac{3 \pi}{7}\right) + 2 \cos \left(\frac{\pi}{7}\right) + 2 \cos \left(\frac{5 \pi}{7}\right) + \cos \left(\frac{\pi}{7}\right) \cos \left(\frac{3 \pi}{7}\right) + \cos \left(\frac{3 \pi}{7}\right) \cos \left(\frac{5 \pi}{7}\right) + \cos \left(\frac{5 \pi}{7}\right) \cos \left(\frac{\pi}{7}\right)}{\left(1 + \cos \left(\frac{5 \pi}{7}\right)\right) \left(1 + \cos \left(\frac{3 \pi}{7}\right)\right) \left(1 + \cos \left(\frac{\pi}{7}\right)\right)}\right] - 3$

$= 2 \left[\frac{3 + 2 \cos \left(\frac{3 \pi}{7}\right) + 2 \cos \left(\frac{\pi}{7}\right) + 2 \cos \left(\frac{5 \pi}{7}\right) + \frac{1}{2} \left(\cos \left(\frac{4 \pi}{7}\right) + \cos \left(\frac{2 \pi}{7}\right) + \cos \left(\frac{8 \pi}{7}\right) + \cos \left(\frac{2 \pi}{7}\right) + \cos \left(\frac{6 \pi}{7}\right) + \cos \left(\frac{4 \pi}{7}\right)\right)}{\left(1 + \cos \left(\frac{5 \pi}{7}\right)\right) \left(1 + \cos \left(\frac{3 \pi}{7}\right)\right) \left(1 + \cos \left(\frac{\pi}{7}\right)\right)}\right] - 3$

$= 2 \left[\frac{3 + 2 \cos \left(\frac{3 \pi}{7}\right) + 2 \cos \left(\frac{\pi}{7}\right) + 2 \cos \left(\frac{5 \pi}{7}\right) + \frac{1}{2} \left(- \cos \left(\frac{3 \pi}{7}\right) - \cos \left(\frac{5 \pi}{7}\right) - \cos \left(\frac{\pi}{7}\right) - \cos \left(\frac{5 \pi}{7}\right) - \cos \left(\frac{\pi}{7}\right) - \cos \left(\frac{3 \pi}{7}\right)\right)}{\left(1 + \cos \left(\frac{5 \pi}{7}\right)\right) \left(1 + \cos \left(\frac{3 \pi}{7}\right)\right) \left(1 + \cos \left(\frac{\pi}{7}\right)\right)}\right] - 3$

$= 2 \left[\frac{3 + \cos \left(\frac{3 \pi}{7}\right) + \cos \left(\frac{\pi}{7}\right) + \cos \left(\frac{5 \pi}{7}\right)}{\left(1 + \cos \left(\frac{5 \pi}{7}\right)\right) \left(1 + \cos \left(\frac{3 \pi}{7}\right)\right) \left(1 + \cos \left(\frac{\pi}{7}\right)\right)}\right] - 3$

$= 2 \left[\frac{3 + \cos \left(\frac{3 \pi}{7}\right) + \cos \left(\frac{\pi}{7}\right) + \cos \left(\frac{5 \pi}{7}\right)}{1 + \cos \left(\frac{3 \pi}{7}\right) + \cos \left(\frac{\pi}{7}\right) + \cos \left(\frac{5 \pi}{7}\right) + \cos \left(\frac{\pi}{7}\right) \cos \left(\frac{3 \pi}{7}\right) + \cos \left(\frac{3 \pi}{7}\right) \cos \left(\frac{5 \pi}{7}\right) + \cos \left(\frac{5 \pi}{7}\right) \cos \left(\frac{\pi}{7}\right) + \cos \left(\frac{5 \pi}{7}\right) \cos \left(\frac{3 \pi}{7}\right) \cos \left(\frac{\pi}{7}\right)}\right] - 3$

$= 2 \left[\frac{3 + \cos \left(\frac{3 \pi}{7}\right) + \cos \left(\frac{\pi}{7}\right) + \cos \left(\frac{5 \pi}{7}\right)}{\left(1 + \cos \left(\frac{3 \pi}{7}\right) + \cos \left(\frac{\pi}{7}\right) + \cos \left(\frac{5 \pi}{7}\right) + \frac{1}{2} \left(- \cos \left(\frac{3 \pi}{7}\right) - \cos \left(\frac{5 \pi}{7}\right) - \cos \left(\frac{\pi}{7}\right) - \cos \left(\frac{5 \pi}{7}\right) - \cos \left(\frac{\pi}{7}\right) - \cos \left(\frac{3 \pi}{7}\right)\right) + \cos \left(\frac{5 \pi}{7}\right) \cos \left(\frac{3 \pi}{7}\right)\right) \cos \left(\frac{\pi}{7}\right)}\right] - 3$

$= \textcolor{red}{2 \left[\frac{3 + \cos \left(\frac{3 \pi}{7}\right) + \cos \left(\frac{\pi}{7}\right) + \cos \left(\frac{5 \pi}{7}\right)}{1 + \cos \left(\frac{5 \pi}{7}\right) \cos \left(\frac{3 \pi}{7}\right) \cos \left(\frac{\pi}{7}\right)}\right] - 3}$

$= 2 \left(\frac{3 + \frac{1}{2}}{1 - \frac{1}{8}}\right) - 3$ [ please see the note below ]

$= \frac{2 \times \frac{7}{2}}{\frac{7}{8}} - 3 = 5$

cos((3pi)/7)+cos(pi/7)+cos((5pi)/7))

=1/(2sin(pi/7))[2sin(pi/7)cos(pi/7)+2sin((3pi)/7)cos(pi/7)+2sin(pi/7)cos((5pi)/7))]

$= \frac{1}{2 \sin \left(\frac{\pi}{7}\right)} \left[\sin \left(\frac{2 \pi}{7}\right) + \sin \left(\frac{4 \pi}{7}\right) - \sin \left(\frac{2 \pi}{7}\right) + \sin \left(\frac{6 \pi}{7}\right) - \sin \left(\frac{4 \pi}{7}\right)\right]$

$= \frac{1}{2 \sin \left(\frac{\pi}{7}\right)} \left[\sin \left(\pi - \frac{\pi}{7}\right)\right]$
$= \frac{1}{2 \sin \left(\frac{\pi}{7}\right)} \cdot \sin \left(\frac{\pi}{7}\right)$

$= \frac{1}{2}$

Again

$\cos \left(\frac{5 \pi}{7}\right) \cos \left(\frac{3 \pi}{7}\right) \cos \left(\frac{\pi}{7}\right)$

$= \frac{1}{2 \sin \left(\frac{\pi}{7}\right)} \left[2 \sin \left(\frac{\pi}{7}\right) \cos \left(\pi - \frac{2 \pi}{7}\right) \cos \left(\pi - \frac{4 \pi}{7}\right) \cos \left(\frac{\pi}{7}\right)\right]$

$= \frac{1}{4 \sin \left(\frac{\pi}{7}\right)} \left[2 \sin \left(\frac{2 \pi}{7}\right) \cos \left(\frac{2 \pi}{7}\right) \cos \left(\frac{4 \pi}{7}\right)\right]$

$= \frac{1}{8 \sin \left(\frac{\pi}{7}\right)} \left[2 \sin \left(\frac{4 \pi}{7}\right) \cos \left(\frac{4 \pi}{7}\right)\right]$

$= \frac{1}{8 \sin \left(\frac{\pi}{7}\right)} \sin \left(\frac{8 \pi}{7}\right)$

$= \frac{1}{8 \sin \left(\frac{\pi}{7}\right)} \sin \left(\pi + \frac{\pi}{7}\right)$

$= - \frac{1}{8 \sin \left(\frac{\pi}{7}\right)} \sin \left(\frac{\pi}{7}\right) = - \frac{1}{8}$