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If tan A= (sin B. Sin C)/(cos B +cos C)Prove that tan A/2= tan B/2. Tan C/2

If tan A= (sin B. Sin C)/(cos B +cos C)Prove that tan A/2= tan B/2. Tan C/2

Grade:11

1 Answers

Ajay
209 Points
4 years ago
Cot A = (COS B +  COS C)/(Sin B . SInC)
 
We know Tan A = 2Tan A/2 /(1-Tan^2 A).
this changes into qunadrtatic equaition for Tan A/2 as follows
Tan^2 A/2 + 2 Tan A/2 Cot A – 1 = 0.
 
Tan A/2 = -COT A ±  SQRT (1 + cot^2A)
 
1 + Cot^2 A = (1+Cos C. Cos B)^2 /(Sin B > Sin C)^2
Sunstituiting and Taking only psoitive sign
 Tan A/2 = (1-CosC)(1-Cos B)/(Sin* B Sinc)
 
           =   (4  Sin^2 C/2* SIn ^2 B/2) /(4 SIn B/2 * Cos B/2 *Cos C/2 * Sin C/2)
           =    TAn B/2 * TAn C/2
       
 

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