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Expand cos(A + B + C). Hence prove that cos A cos B cos C = sin A sin B cos C + sin B sin C cos A + sin C sin A cos B, if A + B + C = π/2

Answer»

Taking A + B = X and C = Y 

We get cos (X + Y) = cos X cos Y – sin X sin Y 

(i.e) cos (A + B + C) = cos (A + B) cos C – sin (A + B) sin C 

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

cos (A + B + C) = cos A cos B cos C – sin A sin B cos C – sin A cos B sin C – cos A sin B sin C 

If (A + B + C) = π/2 then cos (A + B + C) = 0 

⇒ cos A cos B cos C – sin A sin B cos C – sin A cos B sin C – cos A sin B sin C = 0 

⇒ cos A cos B cos C = sin A sin B cos C + sin B sin C cos A + sin C sin A cos B



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