Show that $K(z,t) \equiv \int_{\mathbb{R}^3}{e^{i(z,\xi)}\dfrac{\cos {|\xi|t}}{|\xi|^2}d\xi} = K(0,0,|z|,t)$, $z \in \mathbb{R}^3$ in spherical system

Question

Show that $K(z,t) \equiv \int_{\mathbb{R}^3}{e^{i(z,\xi)}\dfrac{\cos {|\xi|t}}{|\xi|^2}d\xi} = K(0,0,|z|,t)$, $z \in \mathbb{R}^3$ when switched to spherical coordinates. According to my textbook this function depends only on $|z|$ and $t$. I don't see why. After switching to spherical coordinate system,instead of $e^{i(z,\xi)}$ now there is $e^{ir\cos{(\theta)}|z|}$ where $r=|\xi|$ according to the textbook. How is this? Any suggestions?