complex numbers - Logical explanation of Euler's formula

This question is a about (if not proving) at least guessing the Euler's formula.
I don't want the proof using the infinite sums.

We can guess by logic that for example that the equation $x^2+1=\sqrt{x}$ has no real solutions because $x^2=\sqrt{x}$ has 2 solutions $x=0, x=1$ but by adding 1 on the left side, we cancel these 2 solutions, so there are no solutions.

I want to know if there is a way to guess by logic that $e^{ix}=\cos(x)+i\sin(x)$. I guess that the most important here here will be $\frac{d}{dx}e^x=e^x$. And suggestions?

Answer

The power series argument, while simple, is indeed unenlightening. You can easily show that $f(\theta)=\cos\left(\theta\right)+i\sin\left(\theta\right)$ satifies both $f'\left(\theta\right)=i\cdot f\left(\theta\right)$ and $f\left(0\right)=1$, and it looks remarkably similar to one definition of $\gamma(t)=e^{\alpha t}$: the function $\gamma\,\colon\mathbb{C}\rightarrow\mathbb{C}$ that both $\left(e^{\alpha t}\right)'=\alpha e^{\alpha t}$ and $\gamma(0)=1$.