How is $\arctan(\sinh(x * \pi))$ the inverse of $\log(\tan(x)) / \pi$

Question

How is $$\arctan(\sinh(x * \pi))$$ the inverse of $$\frac{\log(\tan(x))}{\pi}$$

What is the relationship between $\log(x)$ and $\sinh(x)$. I guess is what my real question is.

Answer 1

It's not. The inverse of your function $y(x)=\arctan(\sinh(\pi x))$ is $y^{-1}(x)=\dfrac{\sinh^{-1}(\tan y)}\pi$ , where $\sinh^{-1}(t)=\ln\Big(t+\sqrt{t^2+1}\Big)$.