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Prove that cos2A – cos2B + cos2C = 1 – 4 sinA · cosB · sinC

Answer»

L.H.S. = cos 2A – cos2B + cos2C 

= -2 sin(A + B) . sin(A – B) + 1 – 2sin2

= 1 – 2 sinC sin (A – B) – 2 sin2C [∵ sin(A+B)-sinc] 

= 1 – 2 sinC [sin(A – B) + sinC] 

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

= 1 – 2 sinC [2sinA . cosB] 

= 1 – 4 sinA cosB sinC = R.H.S.



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