Question

if sin x+sin y=√3(cos y-cos x) prove that sin3x+sin3y=0​

Answer 1

Given

sinx+siny=√3(cosy−cosx)

2sin(x+y/2)cos(x−y/2)=√3⋅2sin(x+y/2)sin(x−y/2)

⇒sin(x+y/2)cos(x−y/2)−√3sin(x+y/2)sin(x−y/2)=0

⇒sin(x+y/2)[cos(x−y/2)−√3sin(x−y/2)]=0

So

sin(x+y/2)=0

⇒x+y/2=0

Now

sin3x+sin3y

=2sin(3(x+y)/2)cos(3(x−y)/2)

=2sin(3⋅0)cos(3(x−y)/2)

=0

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