Euler's formula $e^{i\pi} = -1$

Question

I need some help to verify this, basically I need to use the rule:

$$\exp(-x) = \frac{1}{\exp(x)}$$

to verify that

$$e^{i\pi} = -1$$

Except I'm not sure how to do it, I mean I know of the trigonometric identity of $$e^{i\pi} = \cos(\pi) + i\sin(\pi)$$, how do we get $-1$?

Answer 1

Note that

• $\cos(\pi)=-1$
• $\sin(\pi)=0$

Then:

$$e^{i\pi} = \cos(\pi) + i\sin(\pi)=-1+i\cdot 0$$