$$4 \sin (\alpha +\beta +\gamma ) \sin (\alpha +\beta -\gamma ) \sin (\alpha -\beta +\gamma ) \sin (-\alpha +\beta +\gamma )=\begin{vmatrix}
1 & \cos 2 \alpha & \cos 2 \beta \\
\cos 2 \alpha & 1 & \cos 2 \gamma \\
\cos 2 \beta & \cos 2 \gamma & 1 \\
\end{vmatrix}$$