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Math Help - Please check proof of identity.

  1. #1
    Member Furyan's Avatar
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    Please check proof of identity.

    Hello

    This is the last identity, in this exercise.

    \dfrac{\cot^2\theta(\sec\theta\ – 1)}{1 + \sin\theta} \equiv \dfrac{\sec^2\theat(1 - \sin\theta)}{1 + \sec\theta}

    I seem to have proved it, but I would appreciate someone checking my work to make sure that I haven’t got there by accident.

    Multiplying the numerator and denominator by 1 - \sin\theta and \sec\theta + 1 gets me to:

    \dfrac{\cot^2\theta(\sec^2\theta\ – 1)(1 - \sin\theta)}{\cos^2\theta(\sec\theta + 1)}

    \cot^2\theta(\sec^2\theta\ – 1) = 1, which gives:

    \dfrac{1 - \sin\theta}{\cos^2\theta(\sec\theta + 1)}

    Multiplying numerator and denominator by \sec^2\theta gives:

    \dfrac{\sec^2\theat(1 - \sin\theta)}{1 + \sec\theta}

    QED?

    Thank you
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  2. #2
    MHF Contributor MarkFL's Avatar
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    Re: Please check proof of identity.

    Looks good to me!
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  3. #3
    MHF Contributor Siron's Avatar
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    Re: Please check proof of identity.

    You can also prove the identity by using:
    \frac{A}{B }=\frac{C}{D} \Leftrightarrow A\cdot D=B \cdot C

    That means we have to prove:
    \cot^2(x)[\sec(x)-1][1+\sec(x)]=\sec^2(x)[1-\sin(x)][1+\sin(x)]
    \Leftrightarrow \cot^2(x)[\sec^2(x)-1]=\sec^2(x)[1-\sin^2(x)]
    \Leftrightarrow \cot^2(x)\left[\frac{1}{\cos^2(x)}-1\right]=\sec^2(x)\cos^2(x)
    \Leftrightarrow \cot^2(x)\left[\frac{1-\cos^2(x)}{\cos^2(x)}\right]=\frac{\cos^2(x)}{\cos^2(x)}
    \Leftrightarrow \cot^2(x)\frac{\sin^2(x)}{\cos^2(x)}=1
    \Leftrightarrow \cot^2(x)\tan^2(x)=1
    \Leftrightarrow 1=1
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  4. #4
    Member Furyan's Avatar
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    Re: Please check proof of identity.

    Hi Siron

    Quote Originally Posted by Siron View Post
    You can also prove the identity by using:
    \frac{A}{B }=\frac{C}{D} \Leftrightarrow A\cdot D=B \cdot C

    That means we have to prove:
    \cot^2(x)[\sec(x)-1][1+\sec(x)]=\sec^2(x)[1-\sin(x)][1+\sin(x)]
    \Leftrightarrow \cot^2(x)[\sec^2(x)-1]=\sec^2(x)[1-\sin^2(x)]
    \Leftrightarrow \cot^2(x)\left[\frac{1}{\cos^2(x)}-1\right]=\sec^2(x)\cos^2(x)
    \Leftrightarrow \cot^2(x)\left[\frac{1-\cos^2(x)}{\cos^2(x)}\right]=\frac{\cos^2(x)}{\cos^2(x)}
    \Leftrightarrow \cot^2(x)\frac{\sin^2(x)}{\cos^2(x)}=1
    \Leftrightarrow \cot^2(x)\tan^2(x)=1
    \Leftrightarrow 1=1
    I really like that method, a lot. It seems more natural to me than trying to manipulate one side into the other. Thank you very much for taking the time to post it. I've got more identities to prove in other chapters and I'll be looking to use your method whenever I can.

    Thanks again. It's enormously helpful, there's nothing like that in any of my books.
    Last edited by Furyan; December 28th 2011 at 04:22 AM. Reason: Addition
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  5. #5
    MHF Contributor Siron's Avatar
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    Re: Please check proof of identity.

    You're welcome!

    Like you see, it can be useful to prove rational identities.
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