In any triangle `A B C` , prove that following: `\ \ asin(A/2)sin((B-C)/2)+bsin(B/2)sin((C-A)/2)+c sin(C/2)sin((A-B)/2)=0.`

Explanation

Consider

`a sin frac{A}{2} sin (frac{B-C}{2})+b sin frac{B}{2} sin (frac{C-A}{2})+c sin frac{C}{2} sin (frac{A-B}{2})`

`=k[sin A sin frac{A}{2} sin (frac{B-C}{2})+sin B sin frac{B}{2} sin (frac{C-A}{2})+sin C sin frac{C}{2} sin (frac{A-B}{2})]`

`=k`

`[`

`sin {pi-(B+C)} sin frac{A}{2} sin (frac{B-C}{2})+sin {pi-(C+A)} sin frac{B}{2} sin (frac{C-A}{2})+sin`

`{pi-({A}+{B})} sin frac{C}{2} sin (frac{A-B}{2})`

`(therefore A+B+C=pi)`

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