# Simplifying an expression: Stuck with it I have to prove that the expression (omega C - 1/(omega L))/(omega C - 1/(omega L) + omega L - 1(omega C)) is equal to 1/(3-( ((omega_r)/(omega))^2 + ((omega)/(omega_r))^2)) where omega_r= 1/(sqrt(LC))

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
$\frac{1}{3-\left(\left(\frac{{\omega }_{r}}{\omega }{\right)}^{2}+\left(\frac{\omega }{{\omega }_{r}}{\right)}^{2}\right)}$
where ${\omega }_{r}=\frac{1}{\sqrt{LC}}$.
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lipovicai1w
The expressions are NOT equal when $\omega =2$ and $L=1$ and $C=1$:
$\frac{\omega C-\frac{1}{\omega L}}{\omega C-\frac{1}{\omega L}+\omega L-\frac{1}{\omega C}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}=\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\frac{2\cdot 1\phantom{\rule{thickmathspace}{0ex}}-\phantom{\rule{thickmathspace}{0ex}}\frac{1}{2\cdot 1}}{2\cdot 1\phantom{\rule{thickmathspace}{0ex}}-\phantom{\rule{thickmathspace}{0ex}}\frac{1}{2\cdot 1}\phantom{\rule{thickmathspace}{0ex}}+\phantom{\rule{thickmathspace}{0ex}}2\cdot 1\phantom{\rule{thickmathspace}{0ex}}-\phantom{\rule{thickmathspace}{0ex}}\frac{1}{2\cdot 1}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}=\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\frac{\frac{3}{2}}{\phantom{\rule{thickmathspace}{0ex}}4-1\phantom{\rule{thickmathspace}{0ex}}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}=\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\frac{1}{2}$
and
$\frac{1}{3\phantom{\rule{thickmathspace}{0ex}}-\phantom{\rule{thickmathspace}{0ex}}\left[{\left(\frac{{\omega }_{r}}{\omega }\right)}^{2}+{\left(\frac{\omega }{{\omega }_{r}}\right)}^{2}\right]}\phantom{\rule{thickmathspace}{0ex}}=\phantom{\rule{thickmathspace}{0ex}}\frac{1}{3\phantom{\rule{thickmathspace}{0ex}}-\phantom{\rule{thickmathspace}{0ex}}\left[{\left(\frac{1}{2}\right)}^{2}+{\left(\frac{2}{1}\right)}^{2}\right]}\phantom{\rule{thickmathspace}{0ex}}=\phantom{\rule{thickmathspace}{0ex}}\frac{1}{\phantom{\rule{thickmathspace}{0ex}}3\phantom{\rule{thickmathspace}{0ex}}-\phantom{\rule{thickmathspace}{0ex}}\frac{1}{4}\phantom{\rule{thickmathspace}{0ex}}-\phantom{\rule{thickmathspace}{0ex}}4\phantom{\rule{thickmathspace}{0ex}}}\phantom{\rule{thickmathspace}{0ex}}=\phantom{\rule{thickmathspace}{0ex}}\frac{1}{\phantom{\rule{thickmathspace}{0ex}}-\frac{5}{4}\phantom{\rule{thickmathspace}{0ex}}}\phantom{\rule{thickmathspace}{0ex}}=\phantom{\rule{thickmathspace}{0ex}}-\frac{4}{5},$
where I've used the fact that $\phantom{\rule{thickmathspace}{0ex}}{\omega }_{r}=\frac{1}{\sqrt{LC}}=\frac{1}{\sqrt{1\cdot 1\phantom{\rule{thickmathspace}{0ex}}}}=1.$