Answer
Please see the figure attached.
Work Step by Step
We have $\rho = \csc \phi $. Write
$\rho = \frac{1}{{\sin \phi }}$
$\rho \sin \phi = 1$
Recall the relations between cylindrical and spherical coordinates can be found using rectangular coordinates $x$, $y$, $z$ as in the following:
$x = r\cos \theta = \rho \sin \phi \cos \theta $
$y = r\sin \theta = \rho \sin \phi \sin \theta $
$z = \rho \cos \phi $
So, $r = \rho \sin \phi $ and $z = \rho \cos \phi $. Whereas $\theta$ is the same in both cylindrical and spherical coordinates.
Thus, $r = \rho \sin \phi = 1$
Since $r = \sqrt {{x^2} + {y^2}} $, so the surface has equation ${x^2} + {y^2} = 1$. This is the equation of a cylinder whose cross-section has radius $1$ centered at the origin.
