The value of Cosπ7Cos2π7Cos10π7−sinπ14sin3π14sin5π14 is

sinπ14sin3π14sin5π14=cos(π2−π14).cos(π2−3π14).cos(π2−5π14) =cos3π7cos2π7cosπ7=−cosπ7cos2π7cos4π7∴ Required value = 2cosπ7cos2π7cos4π7=−14

The value of Cosπ7Cos2π7Cos10π7−sinπ14sin3π14sin5π14 is

sinπ14sin3π14sin5π14=cos(π2−π14).cos(π2−3π14).cos(π2−5π14) =cos3π7cos2π7cosπ7=−cosπ7cos2π7cos4π7∴ Required value = 2cosπ7cos2π7cos4π7=−14

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The value of $C o s \frac{π}{7} C o s \frac{2 π}{7} C o s \frac{10 π}{7} - \sin \frac{π}{14} \sin \frac{3 π}{14} \sin \frac{5 π}{14}$ is

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$- \frac{1}{4}$

$- \frac{1}{2}$

$\frac{1}{2}$

$\frac{1}{4}$

answer is C.

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Detailed Solution

$\sin \frac{π}{14} \sin \frac{3 π}{14} \sin \frac{5 π}{14} = \cos (\frac{π}{2} - \frac{π}{14}) . \cos (\frac{π}{2} - \frac{3 π}{14}) . \cos (\frac{π}{2} - \frac{5 π}{14}) = \cos \frac{3 π}{7} \cos \frac{2 π}{7} \cos \frac{π}{7} = - \cos \frac{π}{7} \cos \frac{2 π}{7} \cos \frac{4 π}{7}$