<s>[INST] Given unit roots $ω$, find $\frac ω{1+ω^2} + \frac{ω^2}{1+ω^4} + \frac{ω^3}{1+ω} + \frac{ω^4}{1+ω^3}$ Let ω be a complex number such that $ω^5=1$ and $ω≠1$. Find
$$\frac ω{1+ω^2} + \frac{ω^2}{1+ω^4} + \frac{ω^3}{1+ω} + \frac{ω^4}{1+ω^3}$$
I've been having trouble with this unit, need help with solving this problem. (Sorry, I don't know how to format it correctly.)
 [/INST] Note that,
$$\frac ω{1+ω^2}\cdot\frac{ω^3}{ω^3} =  \frac{ω^4}{1+ω^3}, \>\>\>\>\>\>\>
\frac{ω^3}{1+ω}\cdot\frac{ω^4}{ω^4} = \frac{ω^2}{1+ω^4}$$
Thus,
$$\begin{aligned} 
& \frac ω{1+ω^2} + \frac{ω^2}{1+ω^4} + \frac{ω^3}{1+ω} + \frac{ω^4}{1+ω^3} \\
= &2\left(\frac ω{1+ω^2} + \frac{ω^3}{1+ω} \right)
=2\cdot \frac{ω(1+ω)+ω^3(1+ω^2)}{(1+ω^2)(1+ω)}
=2\cdot \frac{ω+ω^2+ω^3+1}{1+ω^2+ω+ω^3}=2
\end{aligned}$$
</s>