test

$$
\begin{vmatrix}
1 & -\textrm{cos}\alpha _{1} & -\textrm{cos}\alpha _{3} \\[0mm]
-\textrm{cos}\alpha _{1} & 1 & -\textrm{cos}\alpha _{2} \\[0mm]
-\textrm{cos}\alpha _{3} & -\textrm{cos}\alpha_{2} & 1
\end{vmatrix} =1-\textrm{cos}^{2}\alpha _{1}-\textrm{cos}^{2}\alpha _{2}-\textrm{cos}^{2}\alpha _{3}+2\textrm{cos}\alpha _{1}\textrm{cos}\alpha _{2}\textrm{cos}\alpha _{3} \\　
=-4\textrm{cos}\left( \displaystyle \frac{\alpha _{1}+\alpha _{2}+\alpha _{3 } }{2} \right) \textrm{cos}\left( \displaystyle \frac{-\alpha _{1}+\alpha _{2}+\alpha _{3 } }{2} \right) \textrm{cos}\left( \displaystyle \frac{-\alpha _{2}+\alpha _{1}+\alpha _{3 } }{2} \right) \textrm{cos}\left( \displaystyle \frac{-\alpha _{3}+\alpha _{1}+\alpha _{2 } }{2} \right)
$$

$$
=\begin{cases}
>0 & \Longleftrightarrow & \alpha _{1}+\alpha _{2}+\alpha _{3}>\pi \\[0mm]
=0 & \Longleftrightarrow & \alpha _{1}+\alpha _{2}+\alpha _{3}=\pi \\[0mm]
<0 & \Longleftrightarrow & \alpha _{1}+\alpha _{2}+\alpha _{3}<\pi
\end{cases}
$$