Finding the z-transform of $h[n] = a^ncos(2pi frac{n}{F_s}f_0)$ for $n ≥ 0$ and zero for $n < 0$

So I'm trying to decide whether the cosine part is intended to be plugged in for $z$ or whether it is strictly part of $h[n]$. (the number a lies in the open unit disk)

I mean I was pretty sure it was all part of $h[n]$ but then upon performing the z-transform I get this rational function

$$\frac{1 - a\cos(2\pi\frac{f_0}{F_s})z^{-1}}{1-2a\cos(2\pi\frac{f_0}{F_s})z^{-1} + a^2z^{-2}}$$

The thing is then I'm supposed to evaluate the poles and zeros and if you just ignore the cosine parts you get this really nice rational expression which factors and simplifies down to $\displaystyle\frac{z}{z-a}$.

So that has gotten me thinking that maybe I'm not understanding things correctly and the cosine portion is supposed to be plugged in for $z$ or something. Can anyone clarify this for me?