Good Day, I was trying to solve the below problem:
Simplify $$\sum_{m=1}^{6} \frac{1}{(\sin k)(\sin k - \cos k)}$$ where $k = \theta + \frac{m \pi}{4} \text{and } 0 < \theta < \frac{\pi}{2}. $
I was thinking of decomposing the fraction somehow and getting a telescoping sum or something, but was unable to do so. I am absolutely clueless and there is nothing that I know that simplifies or progresses on the problem. The only thing that I think works is $$\frac{1}{(\sin k)(\sin k - \cos k)} = \frac{1}{\sin^2k-\sin k \cos k} = \frac{2}{1 - \cos 2k-\sin2k}$$ but again I've no idea how to proceed.
Any help would be appreciated. Thanks.