Your title and the problem do not match. Which is it?
Euler's formula states that .
This seems trivial but I am going to ask for verification anyways.
Is it just this simple?
(i) You can define where are the standard and on
(ii) You can define and in strict terms of and which have been already defined
It is a way, but not the only one.The usual way of proving the formulas you have used is to use power series ....
You can also doing this, if you know the theorems:
A) By appealing to certain facts about analytic continuation (namely, you can prove that your function is true on a certain line [namely the imaginary one] and go from there)
B) Use (although for this one I'm less sure that it works) particular theorems about the uniqueness to complex valued initial condition equations.