One value ofcosπ6+i sinπ61/2 + cosπ6-i sinπ611/2 is

cosπ6+i sinπ61/2 + cosπ6-i sinπ61/2 =cosπ12+i sinπ12+cos11π12-i sin11π12 =cosπ12+cos11π12+isinπ12-sin11π12 =cosπ12+cosπ-π12+isinπ12-sinπ-π12 =cosπ12-cosπ12+isinπ12-sinπ12=0+i(0)=0

One value ofcosπ6+i sinπ61/2 + cosπ6-i sinπ611/2 is

cosπ6+i sinπ61/2 + cosπ6-i sinπ61/2 =cosπ12+i sinπ12+cos11π12-i sin11π12 =cosπ12+cos11π12+isinπ12-sin11π12 =cosπ12+cosπ-π12+isinπ12-sinπ-π12 =cosπ12-cosπ12+isinπ12-sinπ12=0+i(0)=0

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One value of
${(\cos \frac{π}{6} + i \sin \frac{π}{6})}^{1 / 2} + {(\cos \frac{π}{6} - i \sin \frac{π}{6})}^{11 / 2}$ is

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answer is B.

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Detailed Solution

$\begin{array}{rcl} {(\cos \frac{π}{6} + i \sin \frac{π}{6})}^{1 / 2} + {(\cos \frac{π}{6} - i \sin \frac{π}{6})}^{1 / 2} \\ = & \cos \frac{π}{12} + i \sin \frac{π}{12} + \cos \frac{11 π}{12} - i \sin \frac{11 π}{12} \\ = & (\cos \frac{π}{12} + \cos \frac{11 π}{12}) + i (\sin \frac{π}{12} - \sin \frac{11 π}{12}) \\ = & (\cos \frac{π}{12} + \cos (π - \frac{π}{12})) + i (\sin \frac{π}{12} - \sin (π - \frac{π}{12})) \\ = & (\cos \frac{π}{12} - \cos \frac{π}{12}) + i (\sin \frac{π}{12} - \sin \frac{π}{12}) = 0 + i (0) = 0 \end{array}$