How to prove Euler’s formula: eiφ=cos(φ)+isin(φ)e^{i\varphi}=\cos(\varphi) +i\sin(\varphi)?

Proof:
Consider the function $f(t) = e^{-it}(\cos t + i \sin t)$ for $t \in \mathbb{R}$. By the product rule
$$\begin{eqnarray} f^{\prime}(t) = e^{-i t}(i \cos t - \sin t) - i e^{-i t}(\cos t + i \sin t) = 0 \end{eqnarray}$$
identically for all $t \in \mathbb{R}$. Hence, $f$ is constant everywhere. Since $f(0) = 1$, it follows that $f(t) = 1$ identically. Therefore, $e^{it} = \cos t + i \sin t$ for all $t \in \mathbb{R}$, as claimed.