I have to evaluate the following finite sum:
$$A = \csc^2\left(\frac{\pi}{9}\right) + \csc^2\left(\frac{2\pi}{9}\right) + \csc^2\left(\frac{4\pi}{9}\right)$$
My work:
Set $u = e^{i\pi/9}$, thus $$A = \frac{-4}{(u - 1/u)^2} + \frac{-4}{(u^2 - 1/u^2)^2} + \frac{-4}{(u^4 - 1/u^4)^2}$$ or $$A = -4\left[\left(\frac{u}{u^2-1}\right)^2 + \left(\frac{u^2}{u^4-1}\right)^2 + \left(\frac{u^4}{u^8-1}\right)^2\right]$$
I’m unsure how to proceed. Help, preferably proceeding with complex exponents will be appreciated.
EDIT: if the value of $\csc^2(\pi/3)$ is know, one can reduce the problem to: $$2A + 2\csc^2(\pi/3) = \frac{80}{3}$$ Using $$\sum_{r=1}^{m-1} \csc^2(\frac{r\pi}{m}) = \frac{m^2-1}{3}$$
