Perpendicular Medians

In $\triangle ABC$, the centroid $G$ satisfies $AG \perp BG$, and $AB = 2$.

1. Find the minimum value of $\dfrac{1}{\tan A} + \dfrac{1}{\tan B}$. 2. Prove that $\dfrac{\tan C}{\tan A} + \dfrac{\tan C}{\tan B}$ is constant.