In a textbook I recently read that

$$ A_1e^{x}+A_2e^{-x}+A_3e^{ix}+A_4e^{-ix}$$

(where $A_n\in\mathbb{C}$, $A_i\neq A_j\;\forall\;i\neq j$) can be rewritten as

$$ A'_1\sin x+A'_2\cos x+A'_3\sinh x+A'_4\cosh x$$ (where $A'_n\in\mathbb{R}$).

Is that true? How can I prove that?