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)`

Explanation

Here, we will use,

`cot (A/2) = sqrt((s(s-a))/((s-b)(s-c))) and sin(A/2) = sqrt(((s-b)(s-c))/(bc))`

Now, `L.H.S. = (cot (A/2) +cot (B/2))(asin^2(B/2) + bsin^2(A/2))`

`=(sqrt((s(s-a))/((s-b)(s-c))) + sqrt((s(s-b))/((s-a)(s-c))))(a((s-a)(s-c))/(ac)) +b((s-b)(s-c))/(bc))`

`=sqrts[(s-a+s-b)/(sqrt((s-a)(s-b)(s-c)))](s-c)/c[s-a+s-b]`

`=sqrts[(c(s-c))/(sqrt((s-a)(s-b)(s-c)))]`

`=csqrt((s(s-c))/((s-a)(s-b)))`

`=c cot(C/2) = R.H.S.`