Let sinx + siny = a and cosx + cosy = b. Then sin(x+y) is equal to

a) ab/(a+b)

b) ab/(a²+b²)

c) 2ab/(a²+b²)

d) ab/2(a²+b²)

sinx + siny = a      

2sin(x+y/2)cos(x-y/2) = a            ......................1                   (by using sum formula)

cosx + cosy = b

2cos(x-y/2)cos(x+y/2) = b                    ..................2

divide both

tan(x+y/2) = a/b               ...........3

tan²@ = sec²@ - 1           &         sec@ = 1/cos@ so

cos(x+y/2) = [b²/a²+b²]^1/2 ......................4

sin(x+y/2) = {1-cos²(x+y/2)}^1/2 = [a²/a²+b²]^1/2.............5

sinx+y = 2sin(x+y/2)cos(x+y/2)                                    (sinA = 2sinA/2cosA/2)

from eq 4 & 5

sinx+y = 2ab/a²+b²

option c is correct