$\cos \alpha \sin (\beta -\gamma )+\cos \beta \sin (\gamma -\alpha )+\cos \gamma \sin (\alpha -\beta )=$

$$\begin{gathered} \cos \alpha \sin (\beta -\gamma )+\cos \beta \sin (\gamma -\alpha )+\cos \gamma \sin (\alpha -\beta ) \\ =\frac{1}{2}\left(2\cos \alpha \sin (\beta -\gamma )+2\cos \beta \sin (\gamma -\alpha )+2\cos \gamma \sin (\alpha -\beta )\right) \\ =\frac{1}{2}\left(\sin \left(\alpha +\beta -\gamma \right)-\sin \left(\alpha -\beta +\gamma \right)+\sin \left(\beta +\gamma -\alpha \right)-\sin \left(\beta -\gamma +\alpha \right)+\sin \left(\gamma +\alpha -\beta \right)-\sin \left(\gamma -\alpha +\beta \right)\right) \\ =0 \end{gathered}$$