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Math Help - Proving an identify with tan in it

  1. #1
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    Proving an identify with tan in it

    It's me again and I've got another problem that I've been working on for two days, and I just can't get started on it!

    Prove the identity (1 - tan^2x ) / (1 + tan^2x) = 1 - 2sin^2x.

    So what I have so far is:

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

    Now what I do next?
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  2. #2
    MHF Contributor MarkFL's Avatar
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    Re: Proving an identify with tan in it

    Consider writing:

    \frac{1-\tan^2x}{\sec^2x}=\frac{1}{\sec^2x}-\frac{\tan^2x}{\sec^2x}

    Now rewrite the two terms, and then apply a Pythagorean identity.
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  3. #3
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    Re: Proving an identify with tan in it

    Ok, I understood what you did so far, but now to 'rewrite the terms' has me totally confused. I know the Pythagorean identities by heart, but so far I don't see anywhere I can apply it. Thanks for the patience!
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  4. #4
    MHF Contributor MarkFL's Avatar
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    Re: Proving an identify with tan in it

    Since we have:

    \sec(x)\equiv\frac{1}{\cos(x)} we may write:

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

    \frac{\tan^2(x)}{\sec^2(x)}=\tan^2(x)\cos^2(x)= \sin^2(x)
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    Re: Proving an identify with tan in it

    Ok, so now the Pythagorean identity is cos^2x + sin^2x = 1.

    What we have so far is cos^2 - sin^2, yes? Now... I feel so dumb and stupid, I must have some mental deficiency but I just can't seem to relate the two. Please show me how, and I'll study it till I get it.
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  6. #6
    MHF Contributor MarkFL's Avatar
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    Re: Proving an identify with tan in it

    Yes, you now have:

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

    Now, use \cos^2(x)=1-\sin^2(x) to write:

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

    And you are done.
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    Re: Proving an identify with tan in it

    Ah, it's just a question of logic, isnt it? I don't know why I have none. Well, just gotta work at it! Thank you very much, I understand it now, but I'll go through this example over and over.

    So, all I need to know is these 2 rules:

    tanx = sinx / cosx
    cos^2x + sin^2x = 1

    And then, just apply it everywhere. Ok.
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  8. #8
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    Re: Proving an identify with tan in it

    Just wanted to say, I went and did another past paper question just now on the same thing, and got the right answer now that I know how. Thanks for helping me understand!
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    Re: Proving an identify with tan in it

    Another update for all my avid readers: just did two more examples, and got them quick and easy! I don't know what happened, but it seems I suddenly acquired the wonderful skill of logic.

    I won't post again, I just wanted to express my thanks for the helpfulness of this board.
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