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: eiφ=cos(φ)+isin(φ)?

Answer

Proof:
Consider the function f(t)=eit(cost+isint) for tR. By the product rule
f(t)=eit(icostsint)ieit(cost+isint)=0
identically for all tR. Hence, f is constant everywhere. Since f(0)=1, it follows that f(t)=1 identically. Therefore, eit=cost+isint for all tR, as claimed.

Attribution
Source : Link , Question Author : Jichao , Answer Author : Community

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