This question is connected to my current research where unexpectedly there arise connections between trigonometric/hyperbolic functions and their inverses.

In short, if we introduce some element $\tau$ and a linear operator "standard part" that has the following property:

$$\operatorname{st}(\tau+y)^x=-x\zeta(1-x,1/2+y)$$

for any real or complex numbers $y$ and $x$, we can find the "standard part" of any power and analytic function of $\tau$.

As such, the following interesting relation arises (among others):

$$\operatorname{st} \frac1\pi\ln \left(\frac{\tau +\frac{z}{\pi }}{\tau -\frac{z}{\pi }}\right)=\tan z$$

There are other similarly striking relations.

It is purely technical and can be derived from the mentioned above power rule. I wonder, whether there were other cases when in some algebraic system one could see logarithms be connected to exponents or trigonometric functions to inverse trigonometric in closed form.