Simplifying an expression: Stuck with it

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 ${\omega}_{r}=\frac{1}{\sqrt{LC}}$.

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 ${\omega}_{r}=\frac{1}{\sqrt{LC}}$.