`Delta ABC` में सिद्ध कीजिए कि `a sin A - b sin B = c sin (A-B)`.

Explanation

`R.H.S. = c sin (A-B)`
`= c.[sin A cos B- cos A sin B]`
`rArr R.H.S. = c[ka cos B - Kb cos A] " " [because (sin A)/(a)=(sin B)/(b)=k]`
`= k [ca.cos B - bc.cos A]`
`=k[ca.((a^(2)+c^(2)-b^(2))/(2ac))-bc.((b^(2)+c^(2)-a^(2))/(2bc))]`
`= k[(a^(2)+c^(2)-b^(2))/(2)-(b^(2)+c^(2)-a^(2))/(2)]`
`= (k)/(2)[a^(2)+c^(2)-b^(2)-b^(2)-c^(2)+a^(2)]=(k)/(2)[2a^(2)-2b^(2)]`
`= ka^(2)-kb^(2)=a.ka-b.kb`
`= a sin A - b sin B " " [because (sin A)/(a)=(sin B)/(b)=k]`
`= L.H.S. " "` इति सिद्धम्