Graphs of the Other Trigonometric Functions · Challenge problem

A horizontal line $y=k$, with $k>0$, meets the graph of $y=\tan x$ at point $A$ and the graph of $y=\cot x$ at point $B$, where both points have $0<x<\dfrac{\pi}{2}$. If $AB=\dfrac{\pi}{6}$, find the sum of all possible values of $k$.