If A,B,C are angles of a triangle, then `2sin(A/2)cosec (B/2)sin(C/2)-sinAcot(B/2)-cosA` is (a)independent of A,B,C (b) function of A,B (c)function of C (d) none of these

Explanation

`2sin(A/2)cosec(B/2)sin(C/2) - sinAcot(B/2) - cosA`

`=2sin(A/2)cosec(B/2)sin((pi-(A+B))/2) - 2sin(A/2)cos(A/2)cos(B/2)/sin(B/2) - cosA`

`=2sin(A/2)cosec(B/2)cos((A+B)/2) - 2sin(A/2)cosec(B/2)cos(A/2)cos(B/2) - cosA`

`=2sin(A/2)cosec(B/2)[cos((A+B)/2) - cos(A/2)cos(B/2)] - cosA`

`=2sin(A/2)cosec(B/2)[cos(A/2)cos(B/2)-sin(A/2)sin(B/2) - cos(A/2)cos(B/2)] - cosA`

`=-2sin^2(A/2) - cosA`

`=-(1-cosA) - cosA`

`=-1`

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