# Simplify an expression

Granger

## Homework Statement

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}}##

## Homework Equations

3. The Attempt at a Solution [/B]

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:

Homework Helper

## Homework Statement

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 $\omega_r= \frac{1}{\sqrt{LC}}$

## Homework Equations

3. The Attempt at a Solution [/B]

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.

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.

Staff Emeritus