(1) I guess $$ \sum_{n=1}^\infty\frac{(-1)^{n}}{n}\cot(\frac{n}{2})\cos(nx)=\frac{x^2}{2}\;\; ;\;\; -\pi\leq x\leq\pi $$ Is it true?
Note. If it is true, then also $$2\sum_{n=1}^\infty\frac{(-1)^{n+1}}{n}\cot(\frac{n}{2})\sin^2(\frac{nx}{2})=\sum_{n=1}^\infty\frac{(-1)^{n+1}}{n}\cot(\frac{n}{2})-\sum_{n=1}^\infty\frac{(-1)^{n+1}}{n}\cot(\frac{n}{2})\cos(nx)=\frac{x^2}{2}\;\; ;\;\; -\pi\leq x\leq\pi $$
(2)
Evaluate the following Fourier series
$$
2\sum_{n=1}^\infty\frac{1}{n}\cot(\frac{n}{2})\sin^2(\frac{nx}{2})=\sum_{n=1}^\infty\frac{1}{n}\cot(\frac{n}{2})-\sum_{n=1}^\infty\frac{1}{n}\cot(\frac{n}{2})\cos(nx)\;\; ;\;\; x\notin 2\pi\mathbb{Z}$$
(compare it to the above conjecture !)