TITLE: $e^{i\theta}$ $=$ $\cos \theta + i \sin \theta$, a definition or theorem?
QUESTION [10 upvotes]: My question is simply whether the well-known formula $e^{i \theta}$ $=$ $\cos \theta$ $+$ $i \sin \theta$ a definition or there is some proof of the result. 
It seems to me that the formula is a definition (as is the case with the definition of $e$ from which the definition of $e^x$ can easily be derived). But if I try to form the definition in the same manner I have to us the definition from Complex Analysis. Is there any way to prove the result without using any ideas from Complex Analysis?

REPLY [3 votes]: I think if one sets up analysis for aiming at economy of definition (rather than pedagogy), it would be natural to introduce the exponential function first (through the differential equation $f'=f$, or by its series) and then define cosine and sine by
$$
  \cos(z)=\frac{\exp(z)+\exp(-z)}2
  \qquad\text{and}\qquad
  \sin(z)=\frac{\exp(z)-\exp(-z)}{2\mathbf i}.
$$
Now your result is a theorem, but a pretty obvious one proved by elementary algebra from the definitions.