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Math Help - Trigo Identities - Proving

  1. #1
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    Trigo Identities - Proving

    Can anyone help me with proving this:
    sin^2 2x - sin^2 x = sin x sin 3x
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  2. #2
    Super Member wingless's Avatar
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    \sin^2 2x - \sin^2 x =\sin x \sin 3x

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

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

    4\sin^2 x \cos^2 x - \sin^2 x =\sin x (\cos 2x \sin x + 2\cos^2 x \sin x)

    \sin^2 x (4\cos^2 x - 1)=\sin^2 x (\cos 2x + 2\cos^2 x)

    4\cos^2 x - \cos^2 x - \sin^2 x = \cos^2 x - \sin^2 x + 2\cos^2 x

    x=x
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  3. #3
    Member Jonboy's Avatar
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    i'm confused on how you got the last three steps:





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  4. #4
    Super Member wingless's Avatar
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    Line 5 is got by factoring sin^2 x from line 4. Then divide both sides by sin^2 x to get line 6. Two sides are equal in line 6, so x=x and this statement is true for all x values (except x=0 because we divided both sides by sin^2 x, so you should check this for x=0 manually)
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  5. #5
    Member Jonboy's Avatar
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    thanks for the clarification. i see now, good job, that was some nice mathematical manipulation.
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  6. #6
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    Help

    Firstly, thanks for your quick response. However, we are taught not to do proving like this... We have to do it something like this:


    LHS: xxxxxxxx
    = xxxxxxxxxx
    = xxxxxxxxxx
    = xxxxxxxxxx
    = xxxxxxxxxx = RHS

    Would you mind doing it in this format so that i can understand better? Thanks!
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  7. #7
    Member Jonboy's Avatar
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    Do you know what LHS and RHS mean?

    LHS: Left Hand Side

    RHS: Right Hand Side

    So put all the the steps to the left of the equals sign.

    LHS: (\sin 2x)^2 - \sin^2 x

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

    = 4\sin^2 x \cos^2 x - \sin^2 x

    = \sin^2 x (4\cos^2 x - 1)

    = 4\cos^2 x - \cos^2 x - \sin^2 x

    RHS: \sin x \sin(x+2x)

    = \sin x (\cos 2x \sin x + \cos x \sin 2x)

    = \sin x (\cos 2x \sin x + 2\cos^2 x \sin x)

    = \sin^2 x (\cos 2x + 2\cos^2 x)

    = \cos^2 x - \sin^2 x + 2\cos^2 x = LHS
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