<s>[INST] Let $\omega$ be a complex number such that $\omega^5 = 1$ and $\omega\neq 1$, find ... Let $\omega$ be a complex number such that $\omega^5 = 1$ and $\omega \neq 1$. Find
$$\frac{\omega}{1 - \omega^2} + \frac{\omega^2}{1 - \omega^4} + \frac{\omega^3}{1 - \omega} + \frac{\omega^4}{1 - \omega^3}$$
I've been having trouble with this unit, need help  on solving this problem.
 [/INST] Hint: using $\omega^5=1$ and bringing all fractions to numerator $1$ gives:
$$
\require{cancel}
\begin{align}
\frac{\omega}{1 - \omega^2} + \frac{\omega^2}{1 - \omega^4} + \frac{\omega^3}{1 - \omega} + \frac{\omega^4}{1 - \omega^3} & = \frac{\omega^4}{\omega^4}\frac{\omega}{1 - \omega^2} + \frac{\omega^3}{\omega^3}\frac{\omega^2}{1 - \omega^4} + \frac{\omega^2}{\omega^2}\frac{\omega^3}{1 - \omega} + \frac{\omega}{\omega}\frac{\omega^4}{1 - \omega^3} = \\
 & = \cancel{\frac{1}{\omega^4-\omega}} + \bcancel{\frac{1}{\omega^3-\omega^2}}+\bcancel{\frac{1}{\omega^2-\omega^3}} + \cancel{\frac{1}{\omega-\omega^4}}
\end{align}
$$
</s>