In ` A B C ,` show that `a^2(s-a)+b^2(s-b)+c^(2)(s-c))=4R (1+r sin (A/2)sin( B/2)sin( C/2))`

Options

A. `4R Delta(cos A+sin B+cos C)`
B. `4R Delta(sin A+sin B+sin C)`
C. `4R Delta(1+4sin.(A)/(2)sin.(B)/(2)sin.(C )/(2))`
D. none of these

Answer

C

Explanation

`L.H.S.=(1)/(2)(a^(2)(b+c-a)+b^(2)(c+a-b)+c^(2)(a+b-c))`
`=(1)/(2)(a(b^(2)+c^(2)-a^(2))+b(c^(2)+a^(2)-b^(2))+c(a^(2)+b^(2)-c^(2)))`
`=(1)/(2)(2abc cos A+2abc cos B + 2abc cos C)`
`=abc(1+4 sin.(A)/(2)sin.(B)/(2)sin.(C )/(2))`
`=4R Delta (1+4sin.(A)/(2)sin.(B)/(2)sin.(C )/(2))`