Prove that : $w=\text{tan}(z)$ map the region $\{-\frac{\pi}{2}<Rez<\frac{\pi}{2} \}$ to the region $w=u+iv$, where two rays $\{ u=0,|v|\geq 1\}$ are excluded.
I know this conformal map can be seen as a composition of following simpler maps:
$\zeta=iz,\qquad\eta=e^{\zeta},\qquad w=-i\frac{\eta^2-1}{\eta^2+1}$
But I got stuck in analysising the last map. Maybe it's still too complicated for me to see a clear picture. Hope someone could help. Thanks in advanced.