what is cosix? how can an angle be imaginary?
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?)