If $f(x)$ and $g(x)$ are real functions with $f(x)+g(x)=\sin x$, then can we say that they each have the form $a\sin(bx+c)+d$?

Question

Let $f(x)$ and $g(x)$ be two real functions. If $f(x) + g(x) = \sin(x)$, then can we confidently say that $f(x)$ and $g(x)$ are functions of the type $a\cdot \sin(b\cdot x+c)+d$? If not, then what are some counter examples?

Edit: not rational functions but real functions

Answer 1

Take $f(x)=e^{x}, g(x)=\sin(x)-e^{x}$