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 $\omega_r= \frac{1}{\sqrt{LC}}$.
My attempt:
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.