I am studying a fake proof of FLT which assumes $\mathbb Q\left[\zeta_p \right]$ is a UFD (that is, that the rationals with the pth root of unity is a UFD). We have covered the proofs for n=2, 3, and 4. The graduate student I am studying under tasked me with proving the following, which will be used in our fake proof:
$$x^p+y^p=\prod_{i=0}^{p-1}\left(x\zeta^{i}_p + y\zeta^{p-i}_p\right),\label{1}\tag{1}$$
for p, an odd prime. I am not quite sure how to go about this. We've already proven that
$$x^p+y^p=\prod_{k=0}^{p-1}\left(x + y\zeta^k_p\right),\label{2}\tag{2}$$
and I understand why this formula results from the definition of $\zeta_p$ (see here).
I've explored the expansion of \eqref{1}, as well as tried to look at perhaps a proof by induction or some method of rewriting \eqref{1} in terms of \eqref{2}, but to no avail. Any hints that might help prove the first statement would be appreciated.
Hint: $$\prod_{i=0}^{p-1}\left(x\zeta^{i}_p + y\zeta^{p-i}_p\right) = \prod_{i=0}^{p-1}\zeta_p^i \cdot \prod_{i=0}^{p-1}\left(x+ y\zeta^{p-2i}_p\right)$$ Then observe that when $i$ varies from $0$ to $p-1$, the numbers $\zeta_p^{p-2i}$ are a permutation of the numbers $\zeta_p^i$.