Question: Find all the points where the tangent line is horizontal and vertical for the polar equation $\displaystyle r=1-2cos{\theta}$

I'm trying to find the tangencies by passing through the cartesian.

So far I have this:

$\displaystyle x=cos\theta-2cos^2\theta $

$\displaystyle y=sin\theta-2cos\theta sin\theta$

$\displaystyle \frac{dx}{d\theta}=-sin{\theta}(1-4cos{\theta})$

$\displaystyle \frac{dy}{d\theta}= cos -2cos2{\theta}$

I'm currently on the horizontal tangency but got stuck/wondering if there is another way:

$\displaystyle cos-2cos2{\theta}=0$

Thanks in advance