The easiest thing to do with complex numbers is to multiply and add them so for more complicated functions it is best to work with power series expansions:
If you replace x by "ix" in the first, you get
But , , and then , etc.
so all of the "odd" terms have "i" while the "even" terms do not. Separating "real" and "imaginary" parts,
or . Replacing x by -x, , since cos(-x)= cos(x) and sin(-x)= -sin(x).
Adding those two equations, so . In particular, if we now replace x by ix, we have .
(Which is why you titled this "hyperbolics", right?)