Mylee Underwood

2022-07-06

Evaluating $\frac{\sum _{k=0}^{6}{\mathrm{csc}}^{2}\left(a+\frac{k\pi }{7}\right)}{7{\mathrm{csc}}^{2}\left(7a\right)}$

treccinair

Expert

The result of the summation is:
$\frac{\sum _{k=0}^{6}{\mathrm{csc}}^{2}\left(a+\frac{k\pi }{7}\right)}{7{\mathrm{csc}}^{2}\left(7a\right)}=7$
The summation, after some calculation, is reduced to:
$\frac{\left(AEF+BDF+CDE\right)+DEF}{7DEF}=7$
where
$A=4-2\sqrt{2}+\left(2\sqrt{2}-2\right)cos\left(\frac{\pi }{7}\right)$,
$B=-4+2\sqrt{2}+\left(2-2\sqrt{2}\right)sin\left(\frac{\pi }{14}\right)$
$C=-4+2\sqrt{2}+\left(-2+2\sqrt{2}\right)sin\left(\frac{3\pi }{14}\right)$,
$D=2\sqrt{2}cos\left(\frac{\pi }{7}\right)+sin\left(\frac{3\pi }{14}\right)+2$,
$E=cos\left(\frac{\pi }{7}\right)-2\sqrt{2}sin\left(\frac{\pi }{14}\right)-2$,
$F=\sqrt{2}sin\left(\frac{3\pi }{14}\right)+sin\left(\frac{\pi }{14}\right)-2$.
By substituting and grouping we get:
$\frac{6+\frac{1}{8}}{7\frac{1}{8}}=7$.

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