I am not sure if the question is a duplicate. The conclusion from the title is clear, for example by viewing the wolframalpha graph at here.
But on the other hand I feel intuitively there should be only 4, since a line intersect with $y=\cos[\theta]$ at 2 points, thus $y=1/2$ intersect with $y=\cos[2\theta]$ at 4 points in $[0,2\pi]$. Since $\cos[2\theta]=1/2$ implies a solution, the number of solutions of $y=\cos[2x]-1/2,x\in [0,2\pi]$ and $r=\cos[2\theta],\theta\in \mathbb{S}^{1}$ must be the same. So why there is such a difference?