Is there a way to prove Euler’s Identity without using trigonometry?

Is there a way to prove Euler’s Identity, \(e^{πi} + 1 = 0\), without using trigonometry?

Hi @RZ923!
To begin with, Euler’s Identity (\(e^{i\pi}+1=0\)) is directly related to complex numbers  a topic that is much broader than just trigonometry. These two topics are connected and can be used almost everywhere! The simplest and most common interpretation of complex numbers is a geometric one: vectors in a coordinate system. This interpretation is directly connected with trigonometry: its general form is Euler's formula: \(e^{ix}=\text{cos }x + i\cdot\text{sin }x\) (Euler’s Identity is Euler's formula for \(x=\pi\)).
Euler’s Identity is not just a simple, short, and nice identity that comes from nowhere  it is a huge topic that covers lots of beautiful and interesting information and takes years to find out. Everything is connected here, with trigonometry especially.

@nastya Thanks