I am working on the following exercise:
Let $f \in C^2(\mathbb{R}^n)$ be a harmonic function. Let further $g(x):=tan(f(x))+e^{-|x|^2}$ be bounded. Show that f is constant.
My approach: To show the above, I want to use the liouville theorem for harmonic functions. Since I know that $f$ is harmonic, I need to show that it is also bounded: Since $g(x):=tan(f(x))+e^{-|x|^2}$ I can rewrite to $arctan(g(x)-e^{-|x|^2})=f(x)$. Because $arctan$ is bounded and equal to $f$ it follows that $f$ is bounded. Applying the liouville theorem shows that $f$ is constant. Question: Is my approach correct?