Let's consider
1)
#\cos A\cos B\cos C-\sin A\sinB\cos C#
#=(\cos A\cos B-\sin A\sinB)\cos C#
#=(\cos(A+B))\cos C#
#=\cos(A+B)\cos C\ ..............(1)#
2)
#\sinA\cos B\sin C+\cos A\sinB\sin C#
#=(\sinA\cos B+\cos A\sinB)\sin C#
#=(\sin(A+B))\sin C#
#=\sin(A+B)\sin C\ ............(2)#
Now, subtracting (2) from (1) we get
#\cos A\cos B\cos C-\sin A\sinB\cos C-(\sinA\cos B\sin C+\cos A\sinB\sin C)=\cos(A+B)\cosC-\sin(A+B)\sin C#
#\cos A\cos B\cos C-\sin A\sinB\cos C-\sinA\cos B\sin C-\cos A\sinB\sin C=\cos(\bar{A+B})\cosC-\sin(\bar{A+B})\sin C#
#=\cos(\bar{A+B}+C)#
#=\cos(A+B+C)#