If omega is a complex cube root of unity,then | vedclass

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(D) We know that $\omega^3 = 1$ and $1 + \omega + \omega^2 = 0$,so $\omega + \omega^2 = -1$. For $\omega^{1234}$,we divide $1234$ by $3$: $1234 = 3

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$\frac{-1}{\sqrt{2}}$

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(D) We know that $\omega^3 = 1$ and $1 + \omega + \omega^2 = 0$,so $\omega + \omega^2 = -1$. For $\omega^{1234}$,we divide $1234$ by $3$: $1234 = 3 	imes 411 + 1$,so $\omega^{1234} = \omega^1 = \omega$. For $\omega^{2021}$,we divide $2021$ by $3$: $2021 = 3 	imes 673 + 2$,so $\omega^{2021} = \omega^2$. Substituting these into the expression: $\cos [(\omega + \omega^2) \pi - \frac{\pi}{4}] = \cos [(-1) \pi - \frac{\pi}{4}] = \cos [-\pi - \frac{\pi}{4}] = \cos [-(\pi + \frac{\pi}{4})] = \cos (\pi + \frac{\pi}{4}) = -\cos \frac{\pi}{4} = -\frac{1}{\sqrt{2}}$.

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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