In ` A B C ,(cot (A/2)+cot(B/2))(asin^2(B/2)+bsin^2(A/2))=` (a) `cotC` (b) c`cotC` (c) `cot(C/2)` (d) c`cot(C/2)`

Options

A. `cot C`
B. `c cot C`
C. `cot.(C)/(2)`
D. `c cot.(C)/(2)`

Answer

D

Explanation

`{cot.(A)/(2) + cot.(B)/(2)} {a sin^(2).(B)/(2) + b sin^(2).(A)/(2)}`
`= {(cos.(C)/(2))/(sin.(A)/(2) sin.(B)/(2))} {a sin^(2).(B)/(2) + b sin^(2).(A)/(2)}`
`= {cos.(C)/(2)}{a (sin.(B)/(2))/(sin.(A)/(2)) + b (sin.(A)/(2))/(sin.(B)/(2))}`
`= sqrt((s(s-c))/(ab)) {a(sqrt((s-a)(s-c))/(ac))/(sqrt(((s-b)(s-c))/(bc))) + b(sqrt((s-b)(s-c))/(bc))/(sqrt(((s-a)(s-c))/(ac)))}`
`= sqrt((s(s-c))/(ab)) {sqrt(((s-a)/(s-b)) ab) + sqrt(((s-b)/(s-a)) ab)}`
`= sqrt(2(s-c)) {(s-a+ s-b)/(sqrt((s-a) (s-b)))}`
`= sqrt(2-(s-c)) {(2s-a-b)/sqrt((s-a) (s-b))}`
`= c sqrt((s(-c))/((s-a)(s-b))) = c cot.(C)/(2)`