There is a variety of ways to work this problem. One way is to first do the following rearranging/rewriting of the equation.

$$\frac{2}{r} = \sin \theta - \frac{1}{{\cos \theta }}\hspace{0.5in} \to \hspace{0.5in}\frac{{2\cos \theta }}{r} = \sin \theta \cos \theta - 1$$

At this point we can multiply everything by ${r^2}$ and do a little rearranging as follows,

$$2r\cos \theta = {r^2}\sin \theta \cos \theta - {r^2}\hspace{0.5in} \to \hspace{0.5in} 2r\cos \theta = \left( {r\sin \theta } \right)\left( {r\cos \theta } \right) - {r^2}$$

We can now use the following formulas to finish this problem.

$$x = r\cos \theta \hspace{0.25in}\hspace{0.25in}y = r\sin \theta \hspace{0.25in}\hspace{0.25in}{r^2} = {x^2} + {y^2}$$

Here is the answer for this problem,

$$\require{bbox} \bbox[2pt,border:1px solid black]{{2x = yx - \left( {{x^2} + {y^2}} \right)}}$$