When does $-\frac{\pi z}{2}\cot(\pi z)+\frac{1}{2}=0$ where $z$ is a complex variable?

Question

Let $z$ be a complex variable. Is there someone who can show me when does :$$-\frac{\pi z}{2}\cot(\pi z)+\frac{1}{2}=0$$

Note: I have tried using trigonometric formulas but it didn't work. Maybe I should write $\cot(z\pi)$ as a series sum, but it will be complicated to set $z$ as a complex variable.

Answer 1

Hint:

Substitute and let $u=\pi z$ therefore we have: $$-\frac{1}{2}(1-u \cot u)=0$$

Therefor, $u \cot u=1$ therefore $\cot u=\frac{1}{u}$ therefore $\tan u=u$ or $\tan {\pi z}=\pi z$.