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Q.

If $ω$ is a complex cube root of unity, then $[(ω^{1234} + ω^{2021}) π - \frac{π}{4}] =$

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a

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

b

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

c

$\frac{1}{2}$

d

$\frac{\sqrt{3}}{2}$

answer is D.

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Detailed Solution

$\begin{array}{rcl} \cos ((w^{1234} + w^{2021}) π - \frac{π}{4}) \\ \cos (({(w^{3})}^{411} w + {(w^{3})}^{673} w^{2}) π - \frac{π}{4}) \\ = & \cos ((w + w^{2}) π - \frac{π}{4}) = \cos (- π - \frac{π}{4}) \\ = & \cos (π + \frac{π}{4}) \\ = & - \cos \frac{π}{4} = \frac{- 1}{\sqrt{2}} \end{array}$

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If ω is a complex cube root of unity, then ω1234+ω2021π-π4=