Let \(\omega\) be a complex number such that \(\omega^5 = 1\) and \(\omega \neq 1\). Compute \(\frac{\omega}{1 - \omega^2} + \frac{\omega^2}{1 - \omega^4} + \frac{\omega^3}{1 - \omega} + \frac{\omega^4}{1 - \omega^3}\)
+118609 \(\theta=\frac{2\pi}{5}\)
\(w=e^{(2k\pi/5)i}\qquad k=1,2,3,4\)
That is a start.
+26367 Let \(\omega\) be a complex number such that \(\omega^5 = 1\) and \(\omega \neq 1\).
Compute \(\dfrac{\omega}{1 - \omega^2} + \dfrac{\omega^2}{1 - \omega^4} + \dfrac{\omega^3}{1 - \omega} + \dfrac{\omega^4}{1 - \omega^3}\)
\(\begin{array}{|rcll|} \hline && \mathbf{ \dfrac{\omega}{1 - \omega^2} + \dfrac{\omega^2}{1 - \omega^4} + \dfrac{\omega^3}{1 - \omega} + \dfrac{\omega^4}{1 - \omega^3} } \quad &| \quad \omega^4 = \dfrac{\omega^5}{\omega}=\dfrac{1}{\omega} \\\\ &=& \dfrac{\omega}{1 - \omega^2} + \dfrac{\omega^2}{1 - \dfrac{1}{\omega}} + \dfrac{\omega^3}{1 - \omega} + \dfrac{\omega^4}{1 - \omega^3} \\\\ &=& \dfrac{\omega}{1 - \omega^2} + \dfrac{\omega^2}{\dfrac{\omega-1}{\omega}} + \dfrac{\omega^3}{1 - \omega} + \dfrac{\omega^4}{1 - \omega^3} \\\\ &=& \dfrac{\omega}{1 - \omega^2} - \dfrac{\omega^3}{1 - \omega} + \dfrac{\omega^3}{1 - \omega} + \dfrac{\omega^4}{1 - \omega^3} \\\\ &=& \dfrac{\omega}{1 - \omega^2} + \dfrac{\omega^4}{1 - \omega^3} \quad &| \quad \omega^3 = \dfrac{\omega^5}{\omega^2}=\dfrac{1}{\omega^2} \\\\ &=& \dfrac{\omega}{1 - \omega^2} + \dfrac{\omega^4}{1 - \dfrac{1}{\omega^2}} \\\\ &=& \dfrac{\omega}{1 - \omega^2} + \dfrac{\omega^4}{\dfrac{\omega^2-1}{\omega^2}} \\\\ &=& \dfrac{\omega}{1 - \omega^2} - \dfrac{\omega^6}{1 - \omega^2} \quad | \quad \omega^6 = \omega^5 \omega=\omega \\\\ &=& \dfrac{\omega}{1 - \omega^2} - \dfrac{\omega}{1 - \omega^2} \\\\ &=& \mathbf{0} \\ \hline \end{array}\)
