We set $r=\cos \theta$ and $r=1-\cos \theta$ equal to each other and solve for $\theta$:
\begin{align*}
\cos \theta &= 1 - \cos \theta \\
2\cos \theta &= 1 \\
\cos \theta &= \frac{1}{2} \\
\theta &= \frac{\pi}{3}, \frac{5\pi}{3}
\end{align*}
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