Question

sin x +sin y = a, cos x +cos y= b then sin(x+y) =​

Answer 1

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)