If $f$ is analytic on an open and simply-connected subset of $\Bbb{C}$, and if $\cos{f}$ is constant, then must $f$ also be constant?

Question

I need help with the following question:

Suppose $f$ is analytic on the open and simply connected space $\Omega\subset\mathbb{C}$. Must $f$ be a constant function if $\cos{f}$ is constant?

To me it seems natural that $f$ must be a constant function but I am not sure how to prove it.

Answer 1

If $\cos\circ f$ is $\{c\}$ and $\omega\in\Bbb C$ is such that $\cos\omega=c$, then the range of $f$ is a subset of$$\left\{\omega+2k\pi\,\middle|\,k\in\Bbb Z\right\}\cup\left\{-\omega+2k\pi\,\middle|\,k\in\Bbb Z\right\}.\tag1$$But $f$ is continuous and $\Omega$ is connected. So, $f(\Omega)$ is a connected subset of $(1)$. Can you take it from here?