Dihedral Angles of a Tetrahedron

In a tetrahedron $S\text{-}ABC$, the base $\triangle ABC$ is equilateral. Let $\alpha,\beta,\gamma$ be the plane angles of the dihedral angles along $AB$, $BC$, $CA$ (i.e., $S\text{-}AB\text{-}C$, $S\text{-}BC\text{-}A$, $S\text{-}CA\text{-}B$), with $\alpha,\beta,\gamma\neq\dfrac{\pi}2$. Determine, with proof, which of the following are true.

1. $\dfrac 1{\tan\alpha}+\dfrac 1{\tan\beta}+\dfrac 1{\tan\gamma}$ may be negative. 2. $\alpha+\beta+\gamma<\dfrac{3\pi}2$. 3. $\alpha+\beta+\gamma>\pi$. 4. $\dfrac 1{\tan\alpha}+\dfrac 1{\tan\beta}+\dfrac 1{\tan\gamma}$ is always positive.