In $\mathrm{aABC},\sum {\mathrm{a}}^3 \cos (\mathrm{B}-\mathrm{C})=$

$$\begin{gathered} \mathrm{G}.\mathrm{E}=8{\mathrm{R}}^3\sum {\sin }^3 \mathrm{Acos}(\mathrm{B}-\mathrm{C})=8{\mathrm{R}}^3\sum {\sin }^2 \mathrm{Asin}(\mathrm{B}+\mathrm{C})\cos (\mathrm{B}-\mathrm{C}) \\ =4{\mathrm{R}}^2\sum {\sin }^2 \mathrm{A}\left[\sin 2\mathrm{B}+\sin 2\mathrm{C}\right] \\ =8R^3\left[{\sin }^2 \mathrm{Asin} \mathrm{B} \cos \mathrm{B}+{\sin }^2 \mathrm{Asin} \mathrm{C} \cos C+{\sin }^2 \mathrm{B} \sin \mathrm{C} \cos \mathrm{C}+{\sin }^2 \mathrm{B} \sin \mathrm{A} \cos \mathrm{A}+{\sin }^2 \mathrm{C} \sin \mathrm{A} \cos \mathrm{A}+{\sin }^2 \mathrm{C} \sin \mathrm{B} \cos \mathrm{B}\right] \\ =8{\mathrm{R}}^3\left[\sin \mathrm{A} \sin \mathrm{B}(\sin \mathrm{A} \cos \mathrm{B}+\cos \mathrm{A} \sin \mathrm{B})+\sin \mathrm{B} \sin \mathrm{C}(\sin \mathrm{B} \cos \mathrm{C}+\cos \mathrm{B} \sin \mathrm{C})+\sin \mathrm{C} \sin \mathrm{A}(\sin \mathrm{A} \cos \mathrm{C}+\cos \mathrm{A} \sin \mathrm{C})\right] \\ =8{\mathrm{R}}^3\left[\sin \mathrm{A} \sin \mathrm{B} \sin (\sin \mathrm{A}+\mathrm{B})+\sin \mathrm{B} \sin \mathrm{C} \sin (\sin \mathrm{B}+\mathrm{C})+\sin \mathrm{C} \sin \mathrm{A} \sin (\mathrm{C}+\mathrm{A})\right] \\ =8{\mathrm{R}}^2\left[3 \sin \mathrm{A} \sin \mathrm{B} \sin \mathrm{C}\right]=3\mathrm{abc} \end{gathered}$$