The original problem I was looking at was showing the range of $f(z)=z^2+\cos(z)$ is all of $\mathbb{C}$. By using Little Picard and Schwarz reflection across the real axis $(f(\bar{z})=\overline{f(z)})$, we can say the range contains $\mathbb{C}-\mathbb{R}$. Since $f$ is even on $\mathbb{R}$, the range must contain $\mathbb{C}-\{0\}$. Showing zero is in the range is proving to be the difficult part.
I have made a couple attempts at it using Rouché's theorem and the intermediate value theorem along the imaginary axis, where $f(iy)=-y^2+\cosh(y)$. If you graph this function, you can clearly see that $f(0)>0$ and $f(2i)<0$, but it's not super easy to see that $f(2i)<0$ without a calculator. This is a problem on a timed qualifying exam, so there should be a reasonably simple/quick way to show this, but I'm clearly just not seeing it.
