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Math Help - prove this identity (with add and subtract formulas)

  1. #1
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    prove this identity (with add and subtract formulas)

    sin (x+y) sin (x-y) = sin^2x - sin^2y

    i tried doing it but i ended up expanding which turned out ugly..and the result was sin^2x cos^2x - cos^2x sin^2x which did not equal the right side.... the add and subtract formulas are the following:

    sin (x+y) = sinx cosy + cosx siny
    sin (x-y) = sinx cosy - cosx siny

    please answer as soon as possible
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  2. #2
    Super Member Deadstar's Avatar
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    (\sin(x) \cos(y) + \cos(x)\sin(y))(\sin(x) \cos(y) - \cos(x)\sin(y))<br />

    = \sin^2(x) \cos^2(y) - \sin^2(y) \cos^2(x)

    Now use \cos^2(y) = 1 - \sin^2(y) and \cos^2(x) = 1 - \sin^2(x)...
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  3. #3
    MHF Contributor harish21's Avatar
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    Quote Originally Posted by magmagod View Post
    sin (x+y) sin (x-y) = sin^2x - sin^2y

    i tried doing it but i ended up expanding which turned out ugly..and the result was sin^2x cos^2x - cos^2x sin^2x which did not equal the right side.... the add and subtract formulas are the following:

    sin (x+y) = sinx cosy + cosx siny
    sin (x-y) = sinx cosy - cosx siny

    please answer as soon as possible
    sin (x+y) sin (x-y) = sin^2x - sin^2y

    L.H.S. = sin (x+y) \times sin (x-y) =  (sinx.cosy + cosx. siny) \times (sinx. cosy - cosx. siny)

     = (sin^{2}x.cos^{2}y) - (sinx. cosy) . (cosx .siny) +  (cosx. siny).(sinx. cosy) - (cos^{2}x.sin^{2}y)

    =  (sin^{2}x.cos^{2}y) - (cos^{2}x.sin^{2}y)

    = [sin^{2}x.(1-sin^{2}y)] - [(1-sin^{2}x).sin^{2}y]

    = sin^{2}x - (sin^{2}x.sin^{2}y) -sin^{2}y + (sin^{2}y.sin^{2}x)

    = sin^{2}x -sin^{2}y

    = R.H.S
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