Let $d$ be a positive integer with $d ot{equiv}2$ mod 4, and let $K=mathbb{Q}({zeta}_{pd})$ for S an odd prime $p$ such that $p{equiv}1$ mod $d$. Let $K_{infty}={igcup}_{n{geq}0}K_n$ be the cyclotomic $mathbb{Z}_p$-extension of $K=K_0$. In this paper, explicit generators for the Tate cohomology group $hat{H}^{-1}$($G_{m,n}$ are given when $d=qr$ is a product of two distinct primes, where $G_{m,n}$ is the Galois group Gal($K_m/K_n$) and $C_m$ is the group of cyclotomic units of $K_m$. This generalizes earlier results when $d=q$ is a prime.