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Math Help - Simplification

  1. #1
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    Simplification

    Hello,

    I am having trouble trying to simplify the following expression;

    cos^2(x) - tan^2(x) / 1 + tan^2(x)


    Help is much appreciated,

    Dranalion
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  2. #2
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    Hi Dranalion

    You can try to apply identity :  1 + \tan^2(x) = sec^2(x)
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  3. #3
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    Given the identity,
    sec^2(x) = 1 + tan^2(x)

    This makes the identity:

    cos^2(x) - tan^2(x)/sec^2(x)


    Which is:

    cos^2(x) - tan^2(x)/(1/cos^2(x))


    How do I further simplify this expression?
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  4. #4
    Member eXist's Avatar
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    Quote Originally Posted by songoku View Post
    Hi Dranalion

    You can try to apply identity :  1 + \tan^2(x) = sec^2(x)

    After you apply this you have:

    \frac{cos^2(x)  -  tan^2(x)}{sec^2(x)} = \frac{cos^2(x)  -  tan^2(x)}{\frac{1}{cos^2(x)}} = (cos^2x  -  tan^2x)(cos^2x) = cos^4x - sin^2x

    Hope this helps
    -Chad
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  5. #5
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    Sorry, the identity is actually the divided term to be subtracted from cos^2(x), as in:

    cos^2(x) - [ tan^2(x)/(1/cos^2(x)) ]
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  6. #6
    Member eXist's Avatar
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    Then you just have this:

    cos^2x - \frac{tan^2x}{sec^2x} = cos^2x - \frac{tan^2x}{\frac{1}{cos^2x}} = cos^x - (tan^2x)(cos^2x) = cos^2x - sin^2x

    -Chad
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  7. #7
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    Thanks!

    If you have cos^2(x) - sin^2(x), can you use the identity:
    sin^2(x) + cos^2(x) = 1 to somehow simplify this further? Or is this as far as it can be simplified?
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  8. #8
    Member eXist's Avatar
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    Careful, we have: cos^2x - sin^2x

    The identity reads: cos^2x + sin^2x = 1

    Ours is not the same. In ours, we have the difference, so we cannot use that identity here.
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  9. #9
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    Hi Dranalion

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