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Math Help - Showing A Identity Holds

  1. #1
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    Showing A Identity Holds

    I have to show that the following identity holds when 1 + cos(2Ө) does not equal 0.

    tan(Ө) = sin(2Ө)/1 + cos(2Ө)

    I think to do this, you'd have to use cos2Ө and sin2Ө's formulas to derive a formula for tan(Ө) or tan(2Ө). It's the (2Ө) thing that throws me off.
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  2. #2
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    Quote Originally Posted by SportfreundeKeaneKent View Post
    I have to show that the following identity holds when 1 + cos(2Ө) does not equal 0.

    tan(Ө) = sin(2Ө)/1 + cos(2Ө)

    I think to do this, you'd have to use cos2Ө and sin2Ө's formulas to derive a formula for tan(Ө) or tan(2Ө). It's the (2Ө) thing that throws me off.
    tan(\theta) = \frac{sin(2\theta)}{1+cos(2\theta)}


    = \frac{2~sin(\theta)~cos(\theta)}{1+cos(2\theta)}


    = \frac{2~sin(\theta)~cos(\theta)}{1+cos^2(\theta)-sin^2(\theta)}


    = \frac{2~sin(\theta)~cos(\theta)}{2cos^2(\theta)}


    = \frac{sin(\theta)}{cos(\theta)}


    = tan(\theta)
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  3. #3
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    Thanks. The third line is tricky.

    I wanna get a forumula for tan(2Ө) in terms of tan(Ө) by using the cos(2Ө) = cos^2(Ө) - sin^2(Ө) and sin(2Ө) = 2sinӨcosӨ formulas.

    Can I use tan(2Ө) = sin2Ө/cos2Ө and go on from there? And how would I get it back to tanӨ?
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  4. #4
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    Hello,

    Quote Originally Posted by SportfreundeKeaneKent View Post
    Thanks. The third line is tricky.

    I wanna get a forumula for tan(2Ө) in terms of tan(Ө) by using the cos(2Ө) = cos^2(Ө) - sin^2(Ө) and sin(2Ө) = 2sinӨcosӨ formulas.

    Can I use tan(2Ө) = sin2Ө/cos2Ө and go on from there? And how would I get it back to tanӨ?
    By dividing if necessary

    \tan 2 \theta=\frac{2 \sin \theta \cos \theta}{\cos^2 \theta-\sin^2 \theta}

    Divide by \cos^2 \theta above and below
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  5. #5
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    I can get

    (2sinӨ/cosӨ)/1-tan^2Ө

    2tanӨ/1-tan^2Ө = tan(2Ө)

    Would that be the formula or could you simplify it more?
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  6. #6
    Moo
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    Quote Originally Posted by SportfreundeKeaneKent View Post
    I can get

    (2sinӨ/cosӨ)/1-tan^2Ө

    2tanӨ/1-tan^2Ө = tan(2Ө)

    Would that be the formula or could you simplify it more?
    It's the formula
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