arctan of a complex number given by a relation

Question

Let $\omega$ be a complex number which satisfies $$ i \omega = \frac{u^2 -1}{u^2 +1} $$ for some suitable complex number $u$. Find the value of $\tan^{-1}{\omega} $

How do I solve this question? I think there is some mistake in this question.

Answer 1

If $u=e^{iz}$, then $$ω=\tan(z)=\frac{\frac{u-u^{-1}}{2i}}{\frac{u+u^{-1}}2}.$$ You should be able to continue.