Answer
The equation may be symmetric with respect to the polar axis, the line $\theta=\frac{\pi}{2}$, and the pole.
See graph.
Work Step by Step
Step 1. To test the symmetry with respect to the polar axis, replace $(r,\theta)$ with $(r,-\theta)$; we have $r^2 = cos(-2\theta)$ or $r^2 = cos(2\theta)$ . Thus the equation is symmetric with respect to the polar axis.
Step 2. To test the symmetry with respect to the line $\theta=\frac{\pi}{2}$, replace $(r,\theta)$ with $(-r,-\theta)$; we have $(-r)^2 = cos(-2\theta)$ or$r^2 = cos(2\theta)$ . Thus the equation is symmetric with respect to the line $\theta=\frac{\pi}{2}$.
Step 3. To test the symmetry with respect to the pole, replace $(r,\theta)$ with $(-r,\theta)$; we have $(-r)^2 = cos(2\theta)$ or $r^2 = cos(2\theta)$. Thus the equation is symmetric with respect to the pole.
Step 4. Using test points with $0\leq\theta\leq2\pi$, we can graph the equation as shown in the figure.
