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how do I prove sinx-siny=2 sin(1/2)(x-y)cos(1/2)(x+y)

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encyce1981 | eNotes Newbie

Posted April 28, 2013 at 5:10 PM via web

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how do I prove sinx-siny=2 sin(1/2)(x-y)cos(1/2)(x+y)

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justaguide | College Teacher | (Level 2) Distinguished Educator

Posted April 28, 2013 at 5:40 PM (Answer #1)

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The identity `sinx-siny=2* sin((x - y)/2)*cos((x+y)/2)` has to be proved.

`sin x - sin y`

= `sin ((x + y + x - y)/2) - sin ((x + y - x + y)/2)`

= `sin ((x + y)/2 + (x - y)/2)` - `sin ((x + y)/2 - (x - y)/2)`

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

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

This proves that `sinx-siny=2* sin((x - y)/2)*cos((x+y)/2)`

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