I saw the prior question on this. Here is my approach (using some of the inequalities from the prior question): the range is either all of $\mathbb C$ or $\mathbb C$ without one point. Now, if this point is $c$, we can then define entire function $g(z) =1/(z^2+\cos(z) -c)$. Using the inequalities in the prior question, by defining the same rectangular paths, we can use max mod to find a bound on $g$. The only piece of this that took a bit more thinking is that on the vertical line segments, the real part of the denominator is $\cosh(t) - c + 4\pi^2n^2+t^2$, which tends to infinity as the boxes get larger for any fixed $t$ even. We can conclude $g$ is constant, a contradiction.
Does this make sense? I was not following the inequality needed on the vertical line segments to use Rouche's theorem in the question so I decided to propose an alternative approach. Please also let me know if you have an idea of what the response in the question is reasoning for the vertical line segments to apply Rouche.
Prove that the range of the entire function $z^2+\cos(z)$ is all of $\mathbb{C}$.
