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Prove that
cos(A+B+C)+cos(A+B+C)+cos(AB+C)+cos(A+BC)sin(A+B+C)+sin(A+B+C)sin(AB+C)+sin(A+BC)=cotB

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Solution

LHS=(cos(A+B+C)+cos(A+B+C))+(cos(AB+C)+cos(A+BC))(sin(A+B+C)+sin(A+B+C))(sin(AB+C)sin(A+BC))
=2cosAcos(B+C)+2cosAcos(BC)2sin(B+C)cosA+2cosAsin(BC)=cos(B+C)+cos(BC)sin(B+C)+sin(BC)=2cosBcosC2sinBcosC=cotB=RHS

Hence Proved

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