A hand waving proof of Euler’s formula starts from the following Taylor series:
Plugging in in this series and separating the real and imaginary parts yields the formula.
Here is a sketch for a more formal proof:
1. Consider the initial value problem ; . We solve it by formally setting
Evidently to satisfy the differential equation we must have and the initial condition gives . It follows by induction that .
2. The solution is now redesignated as .
3. We note that this power series has an infinite radius of convergence as . By the existence and uniqueness theorem of differential equations, it is certain that this is the only solution of the given IVP.
4. The trigonometric functions are now defined by and and it immediately follows that . We thus have Euler’s formula.