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Math Help - Help proving one side equals the other.

  1. #1
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    Help proving one side equals the other.

    The original equation is: " 1/1-cosx - 1/ 1+secx = csc^2x + cot^2x "

    These are the steps i've used so far:

    = 1/1-cosx (1+cosx/1+cosx) - 1/1+secx (1-secx/1-secx) ---multiply by the conjugate

    = 1+cosz/sin^2x + 1-secx/tan^2x

    Thats as far as i've been able to get.

    Any help would be greatly appreciated!
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  2. #2
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    \begin{aligned}<br />
   \frac{1}{1-\cos x}-\frac{1}{1+\sec x}&=\frac{1}{1-\cos x}-\frac{\cos x}{1+\cos x} \\ <br />
 & =\frac{1+\cos x-\cos x+\cos ^{2}x}{1-\cos ^{2}x} \\ <br />
 & =\frac{1}{\sin ^{2}x}+\frac{\cos ^{2}x}{\sin ^{2}x}. <br />
\end{aligned}

    You may conclude now.
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  3. #3
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    Thanks for the quick response!!

    Is there any way that you could expand your work a bit, im having trouble following it. I know that secx = 1/cosx, but how do you get that to where it is cosx/1+cosx, and after that im confused a bit as well.

    Thanks!
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  4. #4
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    No problem.

    We have \frac{1}{1+\sec x}. But you know that \cos x\sec x=1, then you just multiply top & bottom by \cos x, and the result follows.
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  5. #5
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    ok, when you multiply the left side by cosx/cosx, i see how that would give you cosx as the numerator, but wouldnt that leave you with 1+secxcosx as the denominator, how does that eqaul 1+cosx? And after that step to get 1/1-cosx - cosx/1+cosx are you multiplying both sides by their conjugate to get a common denominator so that you can combine them to 1-cos^2x?
    Thanks for all the help, i just learned this a few days ago and am still trying to understand all the transformations and steps involved.
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  6. #6
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    Quote Originally Posted by masteroc View Post

    ok, when you multiply the left side by cosx/cosx, i see how that would give you cosx as the numerator, but wouldnt that leave you with 1+secxcosx as the denominator, how does that eqaul 1+cosx?
    Read post #4. (After multiplication yields \cos x+\cos x\sec x.)

    Quote Originally Posted by masteroc View Post

    And after that step to get 1/1-cosx - cosx/1+cosx are you multiplying both sides by their conjugate to get a common denominator so that you can combine them to 1-cos^2x?
    Yes.
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  7. #7
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    Ah, i got it now!

    Thanks for bearing with me on that. As I said, I just learned this and im still trying to get used to all the transformations and steps!

    Thanks again Krizalid!
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