Let C=cos⁡2π7+cos⁡4π7+cos⁡8π7 and S=sin⁡2π7+sin⁡4π7+sin⁡8π7 then

(C+iS)(C−iS)=α+α2+α4α6+α5+α3⇒C2+S2=1+α6+α4+α+1+α5+α3+α2+1a7=1⇒−122+S2=2+1+α+α2+α3+α4+α5+α6⇒S2=2+0−14=74⇒S=72

Let C=cos⁡2π7+cos⁡4π7+cos⁡8π7 and S=sin⁡2π7+sin⁡4π7+sin⁡8π7 then

(C+iS)(C−iS)=α+α2+α4α6+α5+α3⇒C2+S2=1+α6+α4+α+1+α5+α3+α2+1a7=1⇒−122+S2=2+1+α+α2+α3+α4+α5+α6⇒S2=2+0−14=74⇒S=72

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Let $C = \cos \frac{2 π}{7} + \cos \frac{4 π}{7} + \cos \frac{8 π}{7} and S = \sin \frac{2 π}{7} + \sin \frac{4 π}{7} + \sin \frac{8 π}{7}$ then

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$C = \frac{1}{2}$

$S = \frac{1}{2}$

$C = \frac{\sqrt{7}}{2}$

$S = \frac{\sqrt{7}}{2}$

answer is D.

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Detailed Solution

$\begin{array}{l} (C + iS) (C - iS) = (α + α^{2} + α^{4}) (α^{6} + α^{5} + α^{3}) \\ \Rightarrow C^{2} + S^{2} = 1 + α^{6} + α^{4} + α + 1 + α^{5} + α^{3} + α^{2} + 1 \\ (a^{7} = 1) \\ \Rightarrow {(- \frac{1}{2})}^{2} + S^{2} = 2 + (1 + α + α^{2} + α^{3} + α^{4} + α^{5} + α^{6}) \\ \Rightarrow S^{2} = 2 + 0 - \frac{1}{4} = \frac{7}{4} \Rightarrow S = \frac{\sqrt{7}}{2} \end{array}$