I need to redeem my earlier egregious errors:
The thing to remember with a polar plot of the form $r=f(\theta)$ is that it represents the set $\{f(\theta)(\cos \theta, \sin \theta)\}_{\theta \in \mathbb{R}}$. Thus to solve the problem, we need to determine $\Omega = \{\cos 2\theta(\cos \theta, \sin \theta)\}_{\theta \in \mathbb{R}} \cap \{(1+\cos\theta)(\cos \theta, \sin \theta)\}_{\theta \in \mathbb{R}}$.
So, we look for $\theta, \phi$ that solve the equation: $\cos 2\theta(\cos \theta, \sin \theta) = (1+\cos\phi)(\cos \phi, \sin \phi).$
To simplify life, first look for solutions to $\cos 2\theta = 0$, $(1+\cos\phi)=0$. Setting $\theta = \pi, \phi = \frac{\pi}{2}$ shows that $(0,0) \in \Omega$. This gives solution 1).
Now suppose that either $\theta$ and $\phi$ are such that $\cos 2\theta \neq 0$, $(1+\cos\phi) \neq 0$. Then, since the points are the same, we must have $\phi = \theta + n \pi$, for some integer $n$ (since both $(\cos \theta, \sin \theta)$ and $(\cos \phi, \sin \phi)$ lie on the same line through the origin). In our case, we need only consider $n=0,1$.
Taking $n=0$, we look for solutions to $1+ \cos \theta = \cos 2 \theta$. Since $\cos 2 \theta = 2 (\cos \theta)^2-1$, we have $1+ \cos \theta = 2 (\cos \theta)^2-1$. We can solve this by looking for solutions to $1+x = 2 x^2-1$. Solving the quadratic yields $x = \frac{1 \pm \sqrt{17}}{4}$. Since only one of these points lies in $[-1,1]$ we have $\theta = \pm \arccos \frac{1 - \sqrt{17}}{4}$. This gives solution 2).
Now take $n=1$. Look for solutions to $1+ \cos \theta = -\cos 2 \theta$. Using the previous expansion for $\cos 2 \theta$ we end up with the equation $\cos \theta = -2 (\cos \theta)^2$. As above, we look for solutions to $x = -2x^2$ that lie in $[-1,1]$. The solutions are $x=0$, and $x=-\frac{1}{2}$ which both lie in $[-1,1]$. This gives $\theta = \pm \frac{\pi}{2}$ and $\theta = \pm \frac{\pi}{3}$, which gives solutions 3) and 4).
You can use Octave (or Matlab) to see the plot with:
t = 0:0.1:10*pi; plot((1+cos(t)).*cos(t), (1+cos(t)).*sin(t), cos(2*t).*cos(t), cos(2*t).*sin(t))
