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Prove the identity sin x - sin y = 2 sin( (x - y)/2 ) cos( (x + y)/2 )?

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We have to prove the identity sin x - sin y = 2*sin((x - y)/2)*cos((x + y)/2)

Start with 2*sin((x - y)/2)*cos((x + y)/2), use the rules sin(A + B) = sin A*cos B + cos A*sin B and cos(A - B) = cos A*cos B - sin A*sin B

2*sin((x - y)/2)*cos((x + y)/2)

=> 2*[sin(x/2)*cos(y/2) - cos(x/2)sin(y/2)]*[cos(x/2)cos(y/2) - sin(x/2)sin(y/2)]

=> 2*[(sin(x/2)*cos(y/2))(cos(x/2)cos(y/2)) - (cos(x/2)sin(y/2))(cos(x/2)cos(y/2)) - (sin(x/2)*cos(y/2))(sin(x/2)sin(y/2)) + (cos(x/2)sin(y/2))(sin(x/2)sin(y/2))]

=> 2*sin(x/2)(cos(x/2)(cos(y/2)^2 -...

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