Let $\zeta = \exp(2i\pi/p)$, where $p$ is an odd prime. I need to show that
$$\prod_{\substack{1\leqslant i, j\leqslant p-1\\i\neq j}}(\zeta^i-\zeta^j)=\biggl(\prod_{i=1}^{p-1}\zeta^i\biggr)^{p-2}\biggl(\prod_{k=1}^{p-1}(\zeta^k-1)\biggr)^{p-2},$$ where the left-hand side came from computing a Vandermonde matrix.
All I've managed to do so far is write the left-hand side slightly differently as $ \prod_{i=1}^{p-1}\prod_{j=1, j\neq i}^{p-1} (\zeta^i-\zeta^j)$, but I don't see how to get the two products, especially with the $p-2$ power. I think there must be some properties of $\zeta$ as a complex number which I'm not using fully.
We factor out $\zeta_j$ from each of the terms and rearrange: $$\prod_{i\not=j} (\zeta^i-\zeta^j) = \prod_{i\not=j}\zeta^j\prod_{i \not= j} (\zeta^{i-j}-1) = \Big( \prod_{j=1}^{p-1}\zeta_j \Big)^{p-2} \prod_{i \not= j} (\zeta^{i-j \pmod{p}}-1)$$ The last product equals $\prod_{k=1}^{p-1}(\zeta^k - 1)^{p-2}$ since for each $k$, we have $p-2$ terms where $i-j \pmod{p} = k$