Roots of unity over finite fields

Let $H = \{\omega, \omega^2, \dots, \omega^{n-1}, \omega^n = 1\}$ be the multiplicative subgroup of $\mathbb{Z}_p$ of $n$-th roots of unity, generated by the primitive $n$-th root of unity $\omega$.

I am trying to prove (without any success) the following:

• Prove that $-\omega^i = \omega^{i + n/2}$, for every $i \in {1,2,\dots, n}$, i.e., that the additive inverse of $\omega^i$ is $\omega^{i + n/2}$.

Why is this true? I was trying with the sum $\omega^i + \omega^{i+n/2}$, but did not arrived to anything.

Let $n$ be even.

If $w^n=1$, then $w^{n/2} = -1$ and so $w^{i+n/2} = -w^i$.