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Math Help - proving

  1. #1
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    proving

    can you prove that:

    1/(cos^2 x-sin^2 x)^2 - 4tan^2 x/ (1-tan^2 x)^2 =1

    im kinda stuck right now.
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  2. #2
    MHF Contributor harish21's Avatar
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    Quote Originally Posted by lezah View Post
    can you prove that:

    1/(cos^2 x-sin^2 x)^2 - 4tan^2 x/ (1-tan^2 x)^2 =1

    im kinda stuck right now.
    \frac{1}{(cos^2 x-sin^2 x)^2} - \frac{4tan^2 x}{(1-tan^2 x)^2}

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

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

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

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

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

    now expand (cos^2 x-sin^2 x)^2 as cos^4x+sin^4x -2cos^2x. sin^2x = (sin^2x)^2 + (cos^2x)^2-2cos^2x. sin^2x=(sin^2x+cos^2x)^2-2cos^2x. sin^2x-2cos^2x. sin^2x
    Last edited by harish21; April 20th 2010 at 09:26 PM.
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  3. #3
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    Quote Originally Posted by lezah View Post
    can you prove that:

    1/(cos^2 x-sin^2 x)^2 - 4tan^2 x/ (1-tan^2 x)^2 =1

    im kinda stuck right now.
    cos^2(x) - sin^2(x) = cos(2x)
    2tax(x)/(1-tan^(x)] = tan(2x)
    So the given problem is
    1/cos^2(2x) - tan^2(2x)
    sec^2(2x) - tan^2(2x) = .......?
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