$\begin{aligned} & \left[\frac{51+73 \omega+87 \omega^2}{73+87 \omega+51 \omega^2}+\frac{51+73 \omega+87 \omega^2}{87+51 \omega+73 \omega^2}\right]^{15} \\ & =\left[\begin{array}{c}\frac{51 \omega^2+73 \omega^3+87 \omega^4}{73+87 \omega+51 \omega^2} \times \frac{1}{\omega^2} \\ \quad+\frac{51 \omega+73 \omega^2+87 \omega^3}{87+51 \omega+73 \omega^2} \times \frac{1}{\omega}\end{array}\right]^{15} \\ & =\left[\frac{73+87 \omega+51 \omega^2}{73+87 \omega+51 \omega^2} \times \frac{\omega}{\omega^3}+\frac{87+51 \omega+73 \omega^2}{87+51 \omega+73 \omega^2} \times \frac{\omega^2}{\omega^3}\right]^{15} \\ & =\left(\omega+\omega^2\right)^{15}=(-1)^{15}=-1\end{aligned}$