If sinx +siny =a ,cosx+ cosy =b then sin(x+y) equals how much?

Question

If sinx +siny =a ,cosx+ cosy =b then sin(x+y) equals how much? Its an MCQ question.

Answer 1

 

For this problem, you need the following identities

sin(2x) = 2 sinxcosx

sin(x+y) = sinxcosy + sinycosx

sinx + siny = 2sin[(x+y}/2]cos[(x-y)/2]

 

ab = (sinx + siny)(cosx + cosy)

     = sinxcosx + (sinxcosy + sinycosx) + sinycosy

     = ½sin(2x) + sin(x+y) + ½sin(2y)

     = sin(x+y) + ½[sin(2x) + sin(2y)]

     = sin(x+y) + sin(x+y)cos(x-y)

     = sin(x+y) [1 + cos(x-y)]                (1)

 

a² = (sinx + siny)²   = sinx² + 2sinxsiny  + siny²

b² = (cosx + cosy)² = cosx² + 2cosxcosy + cosy²

a² + b² = 2 + 2 cos(x-y)

            = 2 [1 + cos(x-y)]                    (2)

 

From (1) and (2)

 

sin(x+y) = 2ab/(a² + b²)

 

Answer 2

Excuse me. Delete this answer