How can I interpret$~\tan^{}\left(x\right),\ln\left(\tan^{}\left(x\right)\right)~$with$~0<x<\pi~$? I think banned-inputs exist in this range

The following equation is given in the book.

$$y=\ln\left(\tan^{}\left(x\right)\right)+\sin^{-1}\left(\cos^{}\left(x\right)\right)~~\left(0<x<\pi\right)\tag{1}$$

By the way, this equation is given to let one compute $~\frac{dy}{dx}~$

The current dought for me is$~\tan^{}\left(x\right)~$with$~0<x<\pi~$

The followings are in my head.

$$-\infty<\tan^{}\left(x\right)<+\infty~~\leftrightarrow~~-\frac{\pi}{2}<x<\frac{\pi}{2}$$

So the value range of$~0<x<\pi~$has been really making me panicked.

Moreover, as$~\frac{\pi}{2}<x<\pi~$is held,$~\tan^{}\left(x\right)<0~$is satisfied and$~\ln\left(\tan^{}\left(x\right)\right)~$is completely can't be held.

Misprint?

$\tan(x)$ is not defined for $x=π/2$. And $\ln(\tan(x))$ is not real for $π/2<x<π$, so my guess is that there's a typo and it should be $0<x<π/2$