Entire $f$ and $g$ constant if $e^{f(z)}+e^{g(z)}=1$
Solution 1:
I take it
one semester of graduate complex analysis
means you have heard of Picard's so-called "little" theorem.
The functions $e^{f(z)}$ and $e^{g(z)}$ are entire, and omit two values, hence they are constant. That immediately implies that $f$ and $g$ are constant.