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If A+B+C=180 THEN PROVE THAT COSA÷SINB×SINC+COSB÷SINC×SINA+COSC÷SINA×SINB=2

If A+B+C=180 THEN PROVE THAT COSA÷SINB×SINC+COSB÷SINC×SINA+COSC÷SINA×SINB=2

Grade:10

1 Answers

Arun
25758 Points
4 years ago
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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