Hidden Inequalities

In a triangle $ABC$ , if $\tan \frac{A}{2}$ , $\tan \frac{B}{2}$ , and $\tan \frac{C}{2}$ are in a harmonic progression , then the minimum value of $\cot \frac{B}{2}$ can be expressed as $\sqrt x$ , where $x$ is a square-free integer . Find $x$ .