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If $\omega$ is a complex cube root of unity, then $\cos \left[\left(\omega^{1234}+\omega^{2021}\right) \pi-\frac{\pi}{4}\right]$ is equal to
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$\frac{-1}{\sqrt{2}}$
$\begin{aligned} &\cos \left[\left(\omega^{1234}+\omega^{2021}\right) \pi-\frac{\pi}{4}\right]=\cos \left[\left(\omega+\omega^2\right) \pi-\frac{\pi}{4}\right] \\ & =\cos \left[-\pi-\frac{\pi}{4}\right]=\cos \left(\pi-\frac{\pi}{4}\right)=-\cos \frac{\pi}{4}=-\frac{1}{\sqrt{2}}\end{aligned}$
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