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I will state the problem here:
Let $\omega$ be a complex number such that $\omega^5 = 1$ and $\omega \neq 1$. Compute $\frac{\omega}{1 + \omega^2} + \frac{\omega^2}{1 + \omega^4} + \frac{\omega^3}{1 + \omega} + \frac{\omega^4}{1 + \omega^3}.$
W is obviously a complex number. I've tried arranging and rearranging the terms in a way to further simplify the expression, but I'm having trouble getting anywhere with it. Anyone have ideas on how to proceed and solve the problem?
The first and last terms are equal, as can be seen by multiplying the latter's numerator and denominator by $\omega^2$. A similar treatment, with a factor of $\omega$, shows the middle terms are equal too. So we want to double$$\frac{\omega}{1+\omega^2}+\frac{\omega^3}{1+\omega}=\frac{\omega+\omega^2+\omega^3+\omega^5}{(1+\omega)(1+\omega^2)}=1,$$giving $2$ as the final answer. We actually don't need to exclude $\omega=1$ for this to work.
