I'm trying to get through Exercise 6.2 of the book Spectral Methods in MATLAB by Nick Trefethen. Exercise 6.1 gives the coefficients of the N+1 by N+1 matrix (indexed from $0$ to $N$) by: $$D_{ij} = \frac{1}{a_j} \prod^N_{\substack{k=0\\k\neq i,j}}(x_i-x_k) = \frac{a_i}{a_j(x_i-x_j)} \qquad (i\neq j)$$ $$D_{jj} = \sum^N_{\substack{k=0\\k\neq j}}(x_j-x_k)^{-1}$$ where $x_j$ are the Chebyshev points given by $$x_j = \cos\left(\frac{j\pi}{N}\right) \qquad j = 0,1,\dots,N.$$ I'm trying to derive all the coefficients, but I'm starting with deriving $D_{00}$, which according to the book is given by $$D_{00} = \frac{2N^2+1}{6}.$$ I checked this for a couple different $N$ and it seems to hold. If we insert $j=0$ to the second formula and substituting the formula for the Chebyshev points for $x_j$ and $x_k$, we get the summation $$D_{00} = \sum^N_{k=1} \frac{1}{1-\cos(k\pi/N)}.$$ The question is now, where might I go from here? I've tried a lot of things but haven't gotten anywhere. If anyone has any ideas I would very much appreciate it (and maybe some tips for the other coefficients). Thanks!
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