What is the value of: $\cot^{-1}(\cot(\frac{3\pi}4))$?

Question

What is the value of: $\cot^{-1}(\cot(\frac{3\pi}4))$?

The output given by the WolframAlpha is $\frac{-\pi}4$ and by the Symbolab is $\frac{3\pi}4$.

Which one is correct ?

Answer 1

Both are correct. The cotangent of $\frac{3\pi}{4}$ is $-1$, but the arc-cotangent of $-1$ could be either.

Answer 2

They are both correct, it depends on the branch on which you define the arc-cotangent. Indeed as cotangent is a periodic function you can't have an inverse on its entire domain. But if you restrict it to a fundamental domain (open interval of length $\pi$) then you can define an inverse on that subdomain.

Typically the arc-cotangent on the fundamental domain $(-\pi,0)$ o $(0,\pi)$. In the first case you get the answer of WA, in the second of SY.