\text{Abs}[z]=\sqrt{2+2\text{Cos}\left[\frac{x}{2}\right]}=\sqrt{2\left(1+\text{Cos}\left[2\frac{x}{4}\right]\right)}=\sqrt{2\left(2 \text{Cos}^2\left[\frac{x}{4}\right]\right)}=\sqrt{4\text{Cos}^2\left[\frac{x}{4}\right]}=2\text{|Cos}\left[\frac{x}{4}\right]|