<s>[INST] Find $\frac{\sum_\limits{k=0}^{6}\csc^2\left(x+\frac{k\pi}{7}\right)}{7\csc^2(7x)}$ Find the value of 
$\dfrac{\sum_\limits{k=0}^{6}\csc^2\left(x+\dfrac{k\pi}{7}\right)}{7\csc^2(7x)}$
when $x=\dfrac{\pi}{8}$.
The Hint given is: $n\cot nx=\sum_\limits{k=0}^{n-1}\cot\left(x+\dfrac{k\pi}{n}\right)$
I dont know how it comes nor how to use it
 [/INST] with the help of hint $\displaystyle \sum^{n-1}_{k=0}\cot\left(x+\frac{k\pi}{n}\right) = n\cot(nx)$
differentiate with respect to $x$
$\displaystyle \displaystyle -\sum^{n-1}_{k=0}\csc^2\left(x+\frac{k\pi}{n}\right) = -n^2\csc^2(nx)$
$\displaystyle \sum^{n-1}_{k=0}\csc^2\left(x+\frac{k\pi}{n}\right) = n^2\csc^2(nx)$
put $n=7$ and $\displaystyle x = \frac{\pi}{8}$
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