In ` Delta ABC ` , prove that : ` sin ((B - C)/(3)) = ((b -c)/(a)) cos ((A)/(2))`

In ` Delta ABC ` , prove that : ` sin ((B - C)/(3)) = ((b -c)/(a)) cos

1.	In ` Delta ABC ` , prove that : ` sin ((B - C)/(3)) = ((b -c)/(a)) cos ((A)/(2))`
Answer» In `Delta ABC ` , ` a/(sinA) = b/(sinB)= c/(sin C)" "` (by sine rule) By equal ratio theorem we can write . ` (a) /(sin A) = (b - c)/(sin B - sinC )` ` rArr (a)/(b-c) = (sin A)/(sin B - sin C)` ` rArr (b - c)/(a) = (sin B sin C)/(sin A)` `rArr (b-c)/(a) = (2 sin ((B - C)/(2)). cos ((B + C)/(2)))/(sin A)` ` = (2 sin ((B - C)/(2)). cos ((pi - A)/(2)))/(sinA)` (` because A + B + C = pi ` , sum of all angles of triangle) ` = (2 sin ((B-C)/(2)) . sin ((A)/(2)))/(2 sin ((A)/(2)) . cos ((A)/(2)))` ` rArr (b-c)/(a) = (sin ((B - C)/(2)))/(cos((A)/2))` ` rArr ((b - c)/(a)) . cos ((A)/(2)) = sin ((B - C)/(2)) `