Express 4 sinA cosB cosC as the sum of four sines.

Express 4 sinA cosB cosC as the sum of four sines

4sinAcosBcosC

=2(2sinAcosB)cosC

= 2{ sin(A + B) + cos(A – B)}cosC

=2sin(A + B)cosC + 2cos(A – B)cosC

= sin{ (A + B) + C}  + sin{ (A + B) – C)}  + 2sin{ (A – B) + C}  – sin{ (A – B) – C} 

= sin(A+B+C) + sin(A+B-C) + 2sin(A-B+C) – sin(A-B-C)

∴ 4sinAcosBcosC = sin(A+B+C) + sin(A+B-C) + 2sin(A-B+C) – sin(A-B-C)[Proved]