Question

Prove : cos(A+B+C)= cosA cosB cosC - cosAsinBsinC-sinAcosBsinC- sinAsinBsinC​

Answer 1

Answer:

cosA/(sinBsinC) + cosB/(sinCsinA) + cosC/(sinAsinB)=2  

2 (sinBsinC) = cos ( (B – C)/2) -cos ( (B + C)/2)  

cosA/(sinBsinC) + cosB/(sinCsinA) + cosC/(sinAsinB  

= (sin A cos A + sin B cos B + sin C cos C|) / (sin A sin B sin C)  

= (sin (2A) + sin (2B) + sin (2C) ) / (2 sin A sin B sin C)  

= (2 sin (A + B) cos (A – B) + 2 sin C cos C) / (2 sin A sin B sin C)  

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

since A + B = π – C  

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

= 2 (sin A sin B) / (sin A sin B), by the standard expansion,  

= 2

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