Show that a harmonic function is constant by using the maximum principle

Question

Let $u \in C^2(\mathbb{R}^n)$ harmonic and $f(x)=tan(u(x))-e^{|x|^2}$ bounded above. Show that $u$ is constant. (Hint: use the maximum principle)

When $f$ is bounded above then there must exist an $\delta>0$ such that $u(x) \in (-\frac{\pi}{2} + \delta , \frac{\pi}{2} - \delta) $ (modulo multiplies) otherwise $f$ is not bounded. But I can't show that $u$ is constant. Can someone give me a little hint?

Answer 1

A priori the existence of such $\delta$ is far from clear.

What is clear is that $u(x) \in (-\frac{\pi}{2} + k \pi, \frac{\pi}{2} + k\pi)$ for some $k \in \mathbb{Z}$. Then we can apply Liouville's theorem, even in its simplest case of bounded $u$.