Answer
$r = \frac{cos~2\theta}{cos~\theta}$
This graph is a cissoid with a loop.
We can see this graph below:
Work Step by Step
$r = \frac{cos~2\theta}{cos~\theta}$
When $\theta = 0^{\circ}$, then $r = \frac{cos~0^{\circ}}{cos~0^{\circ}} = 1$
When $\theta = 30^{\circ}$, then $r = \frac{cos~60^{\circ}}{cos~30^{\circ}} = 0.58$
When $\theta = 45^{\circ}$, then $r = \frac{cos~90^{\circ}}{cos~45^{\circ}} = 0$
When $\theta = 60^{\circ}$, then $r = \frac{cos~120^{\circ}}{cos~60^{\circ}} = -1$
When $\theta = 75^{\circ}$, then $r = \frac{cos~150^{\circ}}{cos~75^{\circ}} = -3.3$
When $\theta = 85^{\circ}$, then $r = \frac{cos~170^{\circ}}{cos~85^{\circ}} = -11.3$
When $\theta = 90^{\circ}$, then the expression is undefined.
When $\theta = 95^{\circ}$, then $r = \frac{cos~190^{\circ}}{cos~95^{\circ}} = 11.3$
When $\theta = 120^{\circ}$, then $r = \frac{cos~240^{\circ}}{cos~120^{\circ}} = 1$
When $\theta = 150^{\circ}$, then $r = \frac{cos~300^{\circ}}{cos~150^{\circ}} = -0.58$
When $\theta = 180^{\circ}$, then $r = \frac{cos~360^{\circ}}{cos~180^{\circ}} = -1$
When $\theta = 210^{\circ}$, then $r = \frac{cos~420^{\circ}}{cos~210^{\circ}} = -0.58$
When $\theta = 265^{\circ}$, then $r = \frac{cos~530^{\circ}}{cos~265^{\circ}} = 11.3$
When $\theta = 270^{\circ}$, then the expression is undefined.
When $\theta = 275^{\circ}$, then $r = \frac{cos~550^{\circ}}{cos~275^{\circ}} = -11.3$
When $\theta = 300^{\circ}$, then $r = \frac{cos~600^{\circ}}{cos~300^{\circ}} = -1$
When $\theta = 330^{\circ}$, then $r = \frac{cos~660^{\circ}}{cos~330^{\circ}} = 0.58$
This graph is a cissoid with a loop.
We can see this graph below:
