As the proofs of Theorem 1 in "elementary aspects of the Verlinde formula and of the Harder-Narasimhan-Atiyah-Bott formula" are very terse I could not understand all proofs. The paper can be found at https://people.mpim-bonn.mpg.de/zagier/files/mpim/94-5/fulltext.pdf.
I would appreciate it if someone could explain the proofs of (v), (vi) und (ix). I could also not find the multiplication law for the cotangent function used in (ii). In the proof of (iii) I could not understand how $V_h(k)$ is rewritten, how it is known that $\frac{k}{\tan kx}=\sum_{s=0}^{\infty} \frac{(-4)^sB_{2s}}{(2s)!}n^{2s}x^{2s-1}$, how the standard partial fraction decomposition of $\cot x$ gives the result which follows, how to perform induction to show that $\sin^{-2h}$ is a linear combination of $f_{i}(x)$ and how (iii) is rewritten in the proof for (iv). I could also not undertand how the result of (iv) follows from the proof as the expression from (vi) is nowhere to be found in the proof of (vi).
Also I could not come up with alternative formulas for odd negative power sums of $\sin x$. Some tips would be helpful.
Sorry for being dumb :/
Thanks a lot in Advance!