Question

If $\omega$ is a complex cube root of unity, then find the value of $\left(\frac{a+b \omega+c \omega^{2}}{c+a \omega+b \omega^{2}}+\frac{a+b \omega+c \omega^{2}}{b+c \omega+a \omega^{2}}\right)^{2}$.

Answer 1

Required expression
$=\left[\frac{\omega^{2}\left(a+b \omega+c \omega^{2}\right)}{a \omega^{3}+b \omega^{4}+c \omega^{2}}+\frac{\omega\left(a+b \omega+c \omega^{2}\right)}{a \omega^{3}+b \omega+c \omega^{2}}\right]^{2} $
$=\left[\frac{\omega^{2}\left(a+b \omega+c \omega^{2}\right)}{a+b \omega+c \omega^{2}}+\frac{\omega\left(a+b \omega+c \omega^{2}\right)}{a+b \omega+c \omega^{2}}\right]^{2} $
$=\left(\omega^{2}+\omega\right)^{2}=(-1)^{2}$
$=1$