Euler's formula:

It is one of the most prominent formula which illustrates the relationship between the complex exponential function and the trigonometric functions. It also provides a valid transformation between the Cartesian coordinates and the polar coordinates. Therefore, Euler's formula can be found in many mathematical branches, physics and engineering.
The mathematical representation for Euler's formula is:
$$e^{i\theta}= cos(\theta) + isin(\theta)$$
Where $$e$$ is the base of the natural logarithm, $$i$$ is the imaginary unit and  $$\theta \in \mathbb{C}$$
The notion $$e^{i \theta }$$ where is called the unit complex number.

Proof of Euler's formula:

The Euler's formula derivation is based on Taylor series expansions of the exponential function $$e^{z}$$ and the trigonometric functions $$sin(x)$$ and $$cos(x)$$ where $$z \in \mathbb{C}$$ , $$x \in\mathbb{R}$$.

Beginning with the Taylor series expansion of the exponential function $$e^{z}$$ to have:

$$e^{z}=\sum_{n=0}^{\infty}\frac{z^{n}}{n!}=1+\frac{z}{1!}+\frac{z^{2}}{2!}+\frac{z^{3}}{3!}+...$$

Now, without loss of generality, let $$z=ix$$ to have the following form:

$$e^{ix}=1+\frac{ix}{1!}+\frac{\left (ix\right )^{2} }{2!}+\frac{\left (ix \right )^{3}}{3!}+\frac{\left (ix \right )^{4}}{4!}+\frac{\left (ix \right )^{5}}{5!}+\frac{\left (ix \right )^{6}}{6!}+\frac{\left (ix \right )^{7}}{7!}+\frac{\left (ix \right )^{8}}{8!}+...$$

Expand the powered numerators to have:

$$e^{ix}=1+ix-\frac{x^{2}}{2!}-i \frac{x^{3}}{3!}+\frac{x^{4}}{4!}+i \frac{x^{5}}{5!}-\frac{x^{6}}{6!}-i \frac{x^{7}}{7!}+\frac{x^{8}}{8!}+...$$

Rearrange the right hand side terms to yield:

$$e^{ix}= \left (1-\frac{x^{2}}{2!}+\frac{x^{4}}{4!}-\frac{x^{6}}{6!}+\frac{x^{8}}{8!} - ... \right )+i\left (x-\frac{x^{3}}{3!}+\frac{x^{5}}{5!}-\frac{x^{7}}{7!}+... \right )$$

Thus,
$$e^{ix}= cos(x) + isin(x)$$

Moreover, we called the Euler's formula at $$z=\pi$$; the Euler's Identity, which can be written in the form of $$e^{i \pi}+1=0$$.