Calculating exact trig value with double angle formula leads me to find two values for cot(pi/12) when there is only one.

There are two angles $\theta$ in $[0,2\pi)$ with a cotangent of $\sqrt 3$. One is $\frac \pi 6$ and one is $\frac {7 \pi}6$. This is because adding $\pi$ to the angle changes both the cosine and sine to their negatives, so the signs divide out. Half of the first is $\frac \pi{12}$ and half of the second is $\frac {7 \pi}{12}$. They are both solutions to an angle that is half an angle with cotangent of $\sqrt 3$. You need to choose the right one for your problem.

There are two angles in the fundamental domain $(0,\pi)$ of the cotangent so that $2x=\frac\pi6+k\pi$. These are $\frac\pi{12}$ and $\frac\pi{12}+\frac\pi2=\frac{7\pi}{12}$. Your quadratic equation has roots at the cotangent of both of these angles.