Q:

Prove if $w=\cos{\frac{2\pi}{n}}+i\sin{\frac{2\pi}{n}}$ then the $n$ roots can be written as $1,w,w^2,...,w^{n-1}$

The roots of $z^n=1$ are called root of the unity.
Prove if $w=\cos{\frac{2\pi}{n}}+i\sin{\frac{2\pi}{n}}$ then the $n$ roots can be written as $1,w,w^2,...,w^{n-1}$
My attempt:
Let $z,w\in\mathbb{C}$ such that $w=z^n$ then $w^{1/n}=z$, by hypothesis we know that: $w=\cos{\frac{2\pi}{n}}+i\sin{\frac{2\pi}{n}}$ then,$

A:

By exponential form and Euler's identity we have that
$$1=e^{i(2\pi k)}=\cos (2\pi k)+i\sin(2\pi k)$$
therefore
$$\sqrt[n]1=e^{i\frac{2\pi}n k}=\cos \left(\frac{2\pi}n k\right)+i\sin\left(\frac{2\pi}n k\right)=w^k$$
for $k=0,1,\ldots,n-1$.

