Let $f, g$ and $h$ be entire functions such as the following equation holds: $\sin f(z)+\cos g(z)+e^{h(z)}=0$ Show that $f, g$ and $h$ are constant.
My attempt: If the functions omit $2$ values, Little Picard Theorem shows that they are constant. I have tried to put $\sin f$ and $\cos g$ as exponential, but I didn't get anything interesting.
Is this the right way? Thank you.
The trig identity $$\sin A+\sin B=2\sin\frac{A+B}2\cos\frac{A-B}2$$ is probably helpful.
Set $A=f(z)$, $B=\frac\pi2-g(z)$ to find that the entire function $f(z)-g(z)$ has to avoid values $-\frac\pi2+2k\pi$ etc.