What is cosAcosBcosC-sinAsinBcosC-sinAcosBsinC-cosAsinBsinC?

1 Answer

\cos(A+B+C)

Explanation:

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)