Across a number of independent problem books involving complex numbers, I have seen the following function pop up
$$f(z_1,z_2) = \dfrac{z_1 - z_2}{1+ z_1 z_2 }$$
where $z_1, z_2 \in \mathbb{C}$ and $z_1z_2 \neq -1$.
It appears to have some nice properties, such as
If $z_1$ and $z_2$ lie on the unit circle, then $\Re{\big (f(z_1,z_2)\big )}=0$
I am almost certain that the expression has some significance, otherwise I would not be seeing it across multiple independent sources. Unless it is some random conjured-up expression that just happens to have nice properties.
Question: Is there any significance to this expression? Furthermore, is there any particular reason why it follows a similar structure to the familiar trigonometric identity? $$\tan(x-y) = \dfrac{\tan x - \tan y}{1+ \tan x \tan y}$$
