Showing the first cohomology group of roots of unity in a cyclotomic $\mathbb Z_p$ extension is trivial

Question

Let $F_\infty$ be the cyclotomic $\mathbb Z_p$-extension of $F$ - a number field such that $\mu _p \subset F$. Let $W$ be the set of all roots of unity in $F_\infty$.

I want to prove that $H^1 (\Gamma _n ,W)=0$.

Here $\Gamma _n \cong p^n \mathbb Z_p \subset \mathbb Z_p \cong \Gamma = Gal(F_\infty /F)$. Any hints or references would be appreciated.