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Simplify an expression

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  1. Dec 9, 2017 #1
    1. The problem statement, all variables and given/known data
    I have to prove that the expression

    $$\frac{\omega C - \frac{1}{\omega L}}{\omega C - \frac{1}{\omega L} + \omega L - \frac{1}{\omega C}}$$

    is equal to

    $$\frac{1}{3-( (\frac{\omega_r}{\omega})^2 + (\frac{\omega}{\omega_r})^2)}$$

    where ##\omega_r= \frac{1}{\sqrt{LC}}##

    2. Relevant equations
    3. The attempt at a solution


    What I started to do was to get rid of the denominators in the fraction and put everything together

    $$\frac{\omega^2C^2L-C}{\omega^2C^2L-C+\omega^2CL^2-L}$$

    Then I divided the denominator by the numerator

    $$\frac{1}{1+\frac{\omega^2CL^2-L}{\omega^2C^2L-C}}$$

    And I'm kind of stuck now. Can someone give an hint on how should I proceed next?
    Or is there any easier way to start the proof? I'm just looking for a hint, thanks.
     
    Last edited by a moderator: Dec 9, 2017
  2. jcsd
  3. Dec 9, 2017 #2

    Dick

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    You could start by checking for a typo in the expressions you gave. If you put ##L=C=1## you can easily see that they aren't equal.
     
  4. Dec 9, 2017 #3

    Orodruin

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    First of all, get rid of L and C in favour of ##\omega_r## as soon as possible. It will save you a lot of writing and make your goal clearer. Second, your desired expression has a 3 in the denominator and you have a 1. How can you compensate for this?

    I also suggest working with the reciprocal of your expression. It will save you having to write the expression as a quotient.
     
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