Power sums of ζ^k/(1-ζ^k)² over half the p-th roots

Let $F$ be a field of characteristic zero, let $p$ be a prime with $p \neq 2$, and let $\zeta \in F$ be a primitive $p$-th root of unity. Writing $x_k = \zeta^k/(1-\zeta^k)^2$ for $k$ in the integer interval $[1, p/2]$ (natural-number division, so $k$ runs over $1 \le k \le (p-1)/2$), the theorem asserts the conjunction of three identities in $F$, with $p$ read as an element of $F$ via the canonical map from $\mathbb{N}$: first, $\sum_k x_k = -(p^2-1)/24$; second, $\sum_k x_k^2 = (p^2-1)(p^2+11)/1440$; and third, $\sum_k x_k^3 = -\bigl((p^2-1)(2p^4 + 23p^2 + 191)\bigr)/120960$. The divisions by $24$, $1440$ and $120960$ make sense because $F$ has characteristic zero. The hypotheses that $p$ is prime and odd enter both through the primitivity of $\zeta$ (so that no denominator $1-\zeta^k$ vanishes for $1 \le k \le (p-1)/2$) and through the pairing $k \leftrightarrow p-k$, under which $x_k$ is invariant.

These are the first three power sums, obtained from Newton's identities applied to the roots of $y^p - (y-1)^p$, of the abscissae of the half-kernel of $\mu_p$ on the split nodal cubic $y^2 + xy = x^3$; they are precisely the quantities consumed by Vélu's isogeny formulae, and in particular determine $c_4$ and $c_6$ of the quotient node. They feed the toric branch of the zero-component transport law, being cited by WeierstrassCurve.inZeroComponentAt_veluCoord_iff_of_multiplicative and by WeierstrassCurve.valuation_c4_add_veluTSum_lt_one_of_formal_kernel.