Question

If A + B + C = (3π/2) prove that Cos 2A + Cos 2B + Cos 2C = 1 – 4 SinA.Sin B.Sin C.

Answer 1

A + B + C = 3π/2 ---------(1) 

L.H.S = cos 2A + cos 2B + cos 2C 

= 2 cos (A + B). cos (A – B) + 1 – 2 sin² C 

= 2 cos (270° – C). cos (A – B) – 2 sin² C 

= 1 – 2 sin C [cos (A – B) –2 sin² C 

= 1 – 2 sin C[cos (A – B) + sin C] 

= 1 – 2 sin C [cos (A – B) + sin (270° - bar(A + B))]

= 1 – 2 sin C [cos (A – B) – cos (A + B)]

= 1 – 2 sin C [2 sin A sin B] 

= 1 – 4 sin A sin B sin C