Question

If `sinx+siny=sqrt(3)(cos y-cos x),` then `sin3x+sin3y=` (a)`2sin3x` (b) 0 (c) 1 (d) none of these

Answer 1

`sinx+siny = sqrt3(cosy-cosx)`
`=>2sin((x+y)/2)cos((x-y)/2)= sqrt3(2sin((x+y)/2)sin((x-y)/2)) `
`=>sin((x+y)/2)cos((x-y)/2) - sqrt3sin((x+y)/2)sin((x-y)/2) = 0`
`=>sin((x+y)/2)[cos((x-y)/2) - sqrt3sin((x-y)/2)] = 0`
`=>sin((x+y)/2) = 0 or cos((x-y)/2) - sqrt3sin((x-y)/2) = 0`
`=>x+y = 0 or cot ((x-y)/2) = sqrt3`
`=>x = -y or (x-y)/2 = pi/6`
`=>x= -y or x = y+pi/3`
When `x = -y,`
`sin3x+siny = sin(-3y)+sin3y = sin3y-sin3y= 0`
When, `x = y+pi/3,`
`sin3x+siny = sin3(y+pi/3)+sin3y = sin(pi+3y)+sin3y= -sin3y+sin3y = 0`
`:.` In both cases, ` sin3x+sin3y = 0.`