The value of 2sinπ8sin2π8sin3π8sin5π8sin6π8sin7π8 is

2sinπ8sinπ4sin3π8sinπ−3π8sinπ−2π8sinπ−π8=2sinπ812sin3π8sin3π8sinπ4sinπ8=2sin2π8sin23π81212=2−12222+1222=2−14×2=18

The value of 2sinπ8sin2π8sin3π8sin5π8sin6π8sin7π8 is

2sinπ8sinπ4sin3π8sinπ−3π8sinπ−2π8sinπ−π8=2sinπ812sin3π8sin3π8sinπ4sinπ8=2sin2π8sin23π81212=2−12222+1222=2−14×2=18

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The value of $2 \sin \frac{π}{8} \sin \frac{2 π}{8} \sin \frac{3 π}{8} \sin \frac{5 π}{8} \sin \frac{6 π}{8} \sin \frac{7 π}{8}$ is

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$\frac{1}{4 \sqrt{2}}$

$\frac{1}{8}$

$\frac{1}{8 \sqrt{2}}$

$\frac{1}{4}$

answer is C.

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Detailed Solution

$\begin{array}{l} 2 \sin \frac{π}{8} \sin \frac{π}{4} \sin \frac{3 π}{8} \sin (π - \frac{3 π}{8}) \sin (π - \frac{2 π}{8}) \sin (π - \frac{π}{8}) \\ = 2 \sin \frac{π}{8} \frac{1}{\sqrt{2}} \sin \frac{3 π}{8} \sin \frac{3 π}{8} \sin \frac{π}{4} \sin \frac{π}{8} \\ = 2 \sin^{2} \frac{π}{8} \sin^{2} \frac{3 π}{8} \frac{1}{\sqrt{2}} \frac{1}{\sqrt{2}} \\ = {(\sqrt{\frac{\sqrt{2} - 1}{2 \sqrt{2}}})}^{2} {(\sqrt{\frac{\sqrt{2 + 1}}{2 \sqrt{2}}})}^{2} = \frac{2 - 1}{4 \times 2} = \frac{1}{8} \end{array}$