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

Could you provide a proof of Euler’s formula: $e^{i\varphi}=\cos(\varphi) +i\sin(\varphi)$?

Consider the function $f(t) = e^{-it}(\cos t + i \sin t)$ for $t \in \mathbb{R}$. By the product rule
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.