$\begin{aligned} & \cos \alpha+\cos \beta-\cos \gamma-\cos (\alpha+\beta+\gamma) \\ & =2 \sin \frac{2 \alpha+\beta+\gamma}{2} \sin \frac{\beta+\gamma}{2}+2 \sin \frac{\beta+\gamma}{2} \sin \left(\frac{\gamma-\beta}{2}\right) \\ & =2 \sin \frac{\beta+\gamma}{2}\left[\sin \frac{2 \alpha+\beta+\gamma}{2}+\sin \frac{\gamma-\beta}{2}\right] \\ & =4 \sin \frac{\beta+\gamma}{2} \sin \frac{\alpha+\gamma}{2} \cos \frac{\alpha+\beta}{2} \\ & =4 \cos \frac{\alpha+\beta}{2} \sin \frac{\beta+\gamma}{2} \sin \frac{\gamma+\alpha}{2} .\end{aligned}$