\t(a+b+c)[tan A/2 + tanB/2]
sumer vishwakarma , 9 Years ago
Grade 12
Trigonometry> (a+b+c)[tan A/2 + tanB/2]
1 AnswersLab Bhattacharjee
Last Activity: 9 Years ago
Use ![HINT: $$\dfrac a{\sin A}=\cdots=2R\\ \text{ and } \sin A+\sin B+\sin C=2\sin\dfrac A2\cos\dfrac A2+2\sin\dfrac{B+C}2\cos\dfrac{B-C}2$$ $$\\ \text{ But } \sin\dfrac{B+C}2=\cos\dfrac A2\text{ and }\cos\dfrac{B+C}2=\sin\dfrac A2$$ $$\\ \implies a+b+c=2R[2\cos\dfrac A2\left(\cos\dfrac{B+C}2+\cos\dfrac{B-C}2\right )]=2R[4\cos\dfrac A2\cos\dfrac B2\cos\dfrac C2]$$](http://latex.codecogs.com/gif.latex?HINT%3A%20%24%24%5Cdfrac%20a%7B%5Csin%20A%7D%3D%5Ccdots%3D2R%5C%5C%20%5Ctext%7B%20and%20%7D%20%5Csin%20A+%5Csin%20B+%5Csin%20C%3D2%5Csin%5Cdfrac%20A2%5Ccos%5Cdfrac%20A2+2%5Csin%5Cdfrac%7BB+C%7D2%5Ccos%5Cdfrac%7BB-C%7D2%24%24%20%24%24%5C%5C%20%5Ctext%7B%20But%20%7D%20%5Csin%5Cdfrac%7BB+C%7D2%3D%5Ccos%5Cdfrac%20A2%5Ctext%7B%20and%20%7D%5Ccos%5Cdfrac%7BB+C%7D2%3D%5Csin%5Cdfrac%20A2%24%24%20%24%24%5C%5C%20%5Cimplies%20a+b+c%3D2R%5B2%5Ccos%5Cdfrac%20A2%5Cleft%28%5Ccos%5Cdfrac%7BB+C%7D2+%5Ccos%5Cdfrac%7BB-C%7D2%5Cright%20%29%5D%3D2R%5B4%5Ccos%5Cdfrac%20A2%5Ccos%5Cdfrac%20B2%5Ccos%5Cdfrac%20C2%5D%24%24)
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