What is a proof of Euler's formula $e^{ix}=\cos(x)+i \sin(x)$ that doesn't use differentiation or De Moivre's theorem, if it exists at all? The Taylor series expansions of $e^x$, $\sin(x)$ and $\cos(x)$ are the standard way to prove it, but does there exist an elementary way that doesn't use limits, calculus or De Moivre's theorem?
The number $e$ is transcendental, so it seems impossible to obtain any non-trivial result about it that does not involve infinite processes (limits, derivatives, integrals...). If you want to define the exponential function for irrational arguments using the basic operations, it indeed seems difficult to avoid polynomials or rational functions, and these must be of infinite degree. Finally, to establish the theorem, you need some definition of the trigonometric functions that is not based on the complex exponential. For this you need to resort to well-axiomatized geometry and to establish the equivalence with statements in calculus. In the end, it boils down to computing the arc length along a circle, hence establish $\pi$, another transcendental number.
The relationship between $e^x$ and $\sin(x)$ and $\cos(x)$
You know know the the $\text{exp()}$ function is a special one, and has many interesting properties among them is that expression $ e^{x} = \lim_{n \to \infty} (1+\frac{x}{n})^n $
If you expand this with binomial theorem, and put $n \rightarrow \infty$, you would get the series for $\text{exp()}$
$$ e^{x} = \lim_{n \to \infty} (1+\frac{x}{n})^n $$
$$ e^{x} = 1+x+\frac{x^2}{2}+\frac{x^3}{6}+\frac{x^4}{24}+\frac{x^5}{120}+\frac{x^6}{720}+\frac{x^7}{5040} + \dots \dots $$
$$ e^{x}= \sum_{k=0}^{\infty}\frac{x^k}{k!}$$
From this series, replace $x$ with $xi$, where $i = \sqrt{-1}$ and split the result into real and imaginary section
$$ e^{x} = 1+x+\frac{x^2}{2}+\frac{x^3}{6}+\frac{x^4}{24}+\frac{x^5}{120}+\frac{x^6}{720}+\frac{x^7}{5040} + \dots \dots $$
$$ e^{xi} = 1+xi+\frac{(xi)^2}{2}+\frac{(xi)^3}{6}+\frac{(xi)^4}{24}+\frac{(xi)^5}{120}+\frac{(xi)^6}{720}+\frac{(xi)^7}{5040} + \dots \dots $$
$$ e^{xi} = \{ 1-\frac{x^2}{2}+\frac{x^4}{24}-\frac{x^6}{720}+ \dots \} + \{ x-\frac{x^3}{6}+\frac{x^5}{120}-\frac{x^7}{5040} + \dots \} \cdot i$$
$$\sin(x) = x-\frac{x^3}{6}+\frac{x^5}{120}-\frac{x^7}{5040} + \dots $$
$$\cos(x) = 1-\frac{x^2}{2}+\frac{x^4}{24}-\frac{x^6}{720}+ \dots $$
Therefore $e^{xi} = \cos(x)+i\sin(x)$