Simple proofs of the following identity related to the roots of units I'm looking for simple proofs of the following identity
$$
z_k\Pi_{j\neq k}(z_k-z_j)=n,
$$
where each $z_j$ is the n-th root of the unit, i.e., $z_j=e^{\frac{2\pi i j}{n}}$,$j=1,2,\ldots,n$.
Thanks in advance.

Note that:
$$\prod_{j\neq k} (z-z_j) = \frac{z^n-1}{z-z_k}$$
Take the limit as $z\to z_k$ and you get $\prod_{j\neq k}(z_k-z_j)=nz_k^{n-1}$. Multiply both sides by $z_k$.
More generally, if $f(z)$ is a polynomial with $z_1,\dots,z_n$ all the roots, with possible repetitions, then for any $k$:
$$\prod_{j\neq k} (x_k-x_j) = f'(x_k).$$
