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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

Answer» `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`
`:. 2sin(A/2)cosec(B/2)sin(C/2) - sinAcot(B/2) - cosA = -1`
So, it is independent of `A,B and C`.


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