Elegant proof of the Euler equation

The Euler equation is one of the most elegant equations in mathematics. It reads

$$e^{i\varphi} = \cos\varphi +i\sin\varphi.$$

Its demonstration usually involves the Taylor expansions of the trigonometric functions. Here I reproduce a simple and straightforward demonstration that I have found on a math forum. 

Consider the the function

$$f(\varphi)=e^{-i\varphi}\left(\cos\varphi+i\sin\varphi\right),$$
with $\varphi\in\mathbb{R}$. It is easy to verify that $f^\prime(\varphi)=0$ for all $\varphi\in\mathbb{R}$. For this reason $f(\varphi)$ is a constant function. Since $f(0)=1$, one must have that

$$e^{i\varphi} = \cos\varphi +i\sin\varphi.$$
If you know another neat way of demonstrating the Euler equation, please comment below.