The value of cos⁡2π7+cos⁡4π7+cos⁡6π7+cos⁡7π7 is

cos⁡2π7+cos⁡4π7+cos⁡6π7+cos⁡7π7=12sin⁡π72sin⁡π7cos⁡2π7+2sin⁡π7cos⁡4π7+2sinπ7cos6π7-1=12sin⁡π7sin⁡π73π7−sin⁡π7+sin⁡5π7−sin⁡3π7+sin7π7-sin5π7-1=12sinπ7-sinπ7-1=-12-1=-32.

The value of cos⁡2π7+cos⁡4π7+cos⁡6π7+cos⁡7π7 is

cos⁡2π7+cos⁡4π7+cos⁡6π7+cos⁡7π7=12sin⁡π72sin⁡π7cos⁡2π7+2sin⁡π7cos⁡4π7+2sinπ7cos6π7-1=12sin⁡π7sin⁡π73π7−sin⁡π7+sin⁡5π7−sin⁡3π7+sin7π7-sin5π7-1=12sinπ7-sinπ7-1=-12-1=-32.

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The value of $\cos \frac{2 π}{7} + \cos \frac{4 π}{7} + \cos \frac{6 π}{7} + \cos \frac{7 π}{7}$ is

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answer is D.

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Detailed Solution

$\cos \frac{2 π}{7} + \cos \frac{4 π}{7} + \cos \frac{6 π}{7} + \cos \frac{7 π}{7}$

$\begin{array}{l} = \frac{1}{2 \sin \frac{π}{7}} [2 \sin \frac{π}{7} \cos \frac{2 π}{7} + 2 \sin \frac{π}{7} \cos \frac{4 π}{7} + 2 \sin \frac{π}{7} \cos \frac{6 π}{7}] - 1 \\ = \frac{1}{2 \sin \frac{π}{7}} [\sin \frac{π}{7} \frac{3 π}{7} - \sin \frac{π}{7} + \sin \frac{5 π}{7} - \sin \frac{3 π}{7} + \sin \frac{7 π}{7} - \sin \frac{5 π}{7}] - 1 \end{array}$