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 -...

(The entire section contains 2 answers and 373 words.)

## Unlock This Answer Now

Start your **48-hour free trial** to unlock this answer and thousands more. Enjoy eNotes ad-free and cancel anytime.

Already a member? Log in here.