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Find the value of $\csc^2 \pi/7 + \csc^2 2\pi/7 + \csc^2 3\pi/7$

My try : Converted it into Sin and then tried to apply series formula but failed

amWhy
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Shinobi
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    Hi, welcome to MSE. For a proficient interaction, please take a few minutes to learn type MathJax. https://math.meta.stackexchange.com/questions/5020/mathjax-basic-tutorial-and-quick-reference – xbh Jul 30 '18 at 15:10
  • It's not a duplicate just want to know the answer I was confused whether 3pi/7 and 4pi/7 are same or not in case of cosec – Shinobi Jul 30 '18 at 15:45
  • @user580093 This is the same, 'cause $\csc^2(\pi - x) = \csc^2 (x)$. – xbh Jul 30 '18 at 15:57

2 Answers2

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The same question is answered here with $\csc^2(4\pi/7)$ instead of $\csc^2(3\pi/7)$.

Lod
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I provide one more solution. Judging by the question you asked, I'm assuming your manipulation of $\tan$ and $\cot$ need to reinforce.

First, some preparation.

We all know that $$ \tan (x \pm y) = \frac {\tan (x) \pm \tan (y)} {1 \mp \tan (x)\tan (y)}. $$ Then $$ \cot (x - y) = \frac {\cot (x) \cot (y) + 1} { \cot (y) - \cot (x)}, $$ or $$ \cot (x) \cot (y) = \cot(x-y) (\cot(y) -\cot(x)) - 1 $$ Especially, $$ \cot (2x) = \frac {\cot^2 (x) -1} {2 \cot (x)}, $$ which yields $$ \cot^2(x) - 1 = 2\cot(x)\cot(2x). $$ Also, $$ \csc^2(x) = \frac {\sin^2(x) + \cos^2(x)} {\sin ^2(x)} = 1 + \cot^2(x). $$ Now let's start. Let $x = \pi /7$. $$ P := \csc^2(x) + \csc^2(2x) + \csc^2(4x) = 3 + \cot^2(x) + \cot^2(2x) + \cot^2(3x). $$ By the formula above, $$ P = 6 + 2(\cot(x) \cot(2x) + \cot(2x) \cot (4x) + \cot (4x) \cot (8x)) =: 6 +2Q. $$ Now change the angle: since $\cot(\pi \pm y) = \mp\cot (y)$, we have $$ Q = \cot(x)\cot(2x) - \cot(2x) \cot (3x) - \cot(3x) \cot (x). $$ Now, \begin{align*} Q &= 1+ \cot(x)(\cot(x) - \cot(2x)) - \cot(x) (\cot(2x) -\cot(3x)) - \cot(2x)(\cot(x) - \cot (3x))\\ &= 1+ \cot(x) (\cot(x) -3\cot(2x) + \cot(3x)) + \cot(2x) \cot(3x)\\ &= \cot(x) (\cot(x) -3\cot(2x) + \cot(3x)) + (\cot(2x)-\cot(3x))\cot(x) \\ &= \cot(x)(\cot(x) -2\cot(2x))\\ &= \cot^2(x) - 2\cot(2x) \cot(x)\\ &=1. \end{align*} Therefore $P = 6+2Q=8$.

xbh
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