Answer
Because the equation $\rho = 1 - \cos \phi $ does not depend on $\theta$, we obtain the surface $S$ by rotating this trace about the $z$-axis.
Work Step by Step
We plot the surface $S$ with equation $\rho = 1 - \cos \phi $ and the trace of $S$ in the $xz$-plane (red curve in the figure) using a computer algebra system. From the figure we see that the surface $S$ is rotationally symmetric with respect to the $z$-axis, therefore rotating the trace of $S$ in the $xz$-plane generates the surface. The fact that the surface $S$ is rotationally symmetric with respect to the $z$-axis is because the equation $\rho = 1 - \cos \phi $ does not depend on $\theta$.
