Prove Euler's identity using Euler's formula.

e^ix = cos x + i sin x

First list all the terms out.

e^ix = 1 + ix/1! + (ix) ^2/2! + (ix) ^3/3! ...

Then, we can expand them.

e^ix = 1 + ix/1! + i^2x^2/2! + i^3x^3/3! ...

Then, we can use the rules of raising i to a power.

e^ix = 1 + ix - x^2/2! - ix^3/3! ...

Then, we can sort all the real and imaginary terms.

e^ix = (1 - x^2/2! ...) + i (x - x^3/3! ...)

We can simplify this.

e^ix = cos x + i sin x

This is Euler's Formula.

What happens if we put in pi?

x = pi

e^i*pi = cos (pi) + i sin (pi)

cos (pi) = - 1

i sin (pi) = 0

e^i*pi = - 1 OR e^i*pi + 1 = 0

That is Euler's identity.