prove that |1 cosA sinA||1 cosB sinB| |1 cosC sinC| = 4 sin(A-B/2) sin(B-C/2) sin(C-A/2) - Math - Determinants

Question

prove that |1 cosA sinA||1 cosB sinB| |1 cosC sinC| = 4 sin(A-B/2)
sin(B-C/2) sin(C-A/2)

Ashwini Kumar · Accepted Answer

LHS=1cosAsinA1cosBsinB1cosCsinC=1cosAsinA0cosB-cosAsinB-sinA0cosC-cosAsinC-sinA         R2→R2-R1 and R3→R3-R1=1cosB-cosAsinC-sinA-sinB-sinAcosC-cosA=2sinA+B2sinA-B2×2sinC-A2cosC+A2-2sinB-A2cosB+A2×2sinA+C2sinA-C2=4sinA+B2sinA-B2sinC-A2cosC+A2-4sinA-B2cosA+B2sinC+A2sinC-A2=4sinA-B2sinC-A2sinA+B2cosC+A2-cosA+B2sinC+A2=4sinA-B2sinC-A2×sinA+B2-C+A2        sin X cos Y-cos X sin Y=sinX-Y=4sinA-B2sinC-A2sinB-C2=4sinA-B2sinB-C2sinC-A2=RHS