Your equation is equivalent to [itex]\rho[/itex] = cos([itex]\phi[/itex])/(1 - cos2([itex]\phi[/itex]).
Your relevant equations show the converstion from spherical to Cartesian coordinates. Do you know the conversions going in the other direction?
One trick that will be helpful is to multiply both sides by rho. This potentially adds a point (rho = 0) that might not be a solution of your original equation, so you should check whether this is already a solution of your equation.
Look at the equations for x, y, and z. Note that they're all of the form rho times some combination of sines and cosines. So the first thing you should do is rewrite everything in terms of sines and cosines. Next, rearrange terms so the trig functions multiply rho. If needed, do as Mark suggested, and multiply both sides by rho. Then try to identify what cartesian coordinates the various products are equal to.