What is cosAcosBcosC-sinAsinBcosC-sinAcosBsinC-cosAsinBsinC?

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Answer 1 · 2018-07-12T21:22:45

#\cos(A+B+C)#

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)#