Write combination of exponentials as combination of $\sinh$ and $\cosh$

Question

Consider a combination of increasing and decreasing complex functions

$$A e^{kx}+B e^{-kx}$$

Can this be always rewritten as a combination of $\sinh(kx)$ and $\cosh(kx)$ functions? i.e.

$$A e^{kx}+B e^{-kx}=C \cosh(kx)+D\sinh(kx)$$ But what are $C$ and $D$?

Answer 1

Yes, since $$ e^{\pm k x} = \cosh{kx} \pm \sinh{kx}, $$ by solving the definitions of the hyperbolic functions for $e^{\pm kx}$. So $$ C=A+B, \quad D=A-B. $$