= Solution
The geometric sum gives
$$
\frac{1-\zeta_p^i}{1-\zeta_p}=1+\zeta_p+\cdots+\zeta_p^{i-1}\equiv i\pmod{\pi_K}.
$$
Since $\Phi_p(1)=p$, its factorization at the nonidentity $p$th <roots of unity> yields
$$
p=\prod_{i=1}^{p-1}(1-\zeta_p^i)=\pi_K^{p-1}w,\qquad w=\prod_{i=1}^{p-1}\frac{1-\zeta_p^i}{1-\zeta_p}.
$$
Each factor is a <unit>, and <Wilson theorem> gives $w\equiv(p-1)!\equiv-1\pmod{\pi_K}$. Set $u=-w^{-1}$. Then $u$ is a <principal unit> and
$$
\boxed{\pi_K^{p-1}=-p u,\qquad u\in1+\pi_K\mathcal O_K.}
$$
The minus sign comes from the product of the nonzero elements of the <residue field>, not from an arbitrary choice of <uniformizer>.
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