Thank you for reporting, we will resolve it shortly
Q.
If $\omega$ is a complex cube root of unity, then $\sin \left\{(\omega^{10}+\omega^{23})\pi-\frac{\pi}{4}\right\}$ is equal to
Complex Numbers and Quadratic Equations
Solution:
Since, $\omega$ is a cube root of unity
$\therefore sin\left\{\left(\omega^{10}+\omega^{23}\right)\pi-\frac{\pi}{4}\right\}$
$=sin\left\{\left(\omega+\omega^{2}\right)\pi-\frac{\pi}{4}\right\}$
$= sin \left(-\pi-\frac{\pi}{4}\right)\quad\left(\because 1+\omega+\omega^{2}=0\right)$
$=-sin \left(\pi+\frac{\pi}{4}\right)=\frac{1}{\sqrt{2}}$