Question

If sinX+sinY=a and cosX+cosY=b
Then prove that sin(x+y)=2ab/(a2+b2)

Answer 1

Answer:

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

Step-by-step explanation:

hope it helps you