$(1 + \cos \frac{π}{8}) (1 + \cos \frac{3 π}{8}) (1 + \cos \frac{5 π}{8}) (1 + \cos \frac{7 π}{8}) =$

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$\frac{1}{8}$

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

$\frac{π}{8}$

$\frac{1}{2}$

answer is C.

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Detailed Solution

We have,
$(1 + \cos \frac{π}{8}) (1 + \cos \frac{3 π}{8}) (1 + \cos \frac{5 π}{8}) (1 + \cos \frac{7 π}{8}) (1 + \cos \frac{π}{8}) (1 + \cos \frac{3 π}{8}) (1 - \cos \frac{3 π}{8}) (1 - \cos \frac{π}{8}) [∵ \cos (π - θ) = \cos θ] = (1 = \cos^{2} \frac{π}{8}) (1 - \cos^{2} \frac{3 π}{8}) = \sin^{2} \frac{π}{8} \sin^{2} \frac{3 π}{8} = \sin^{2} \frac{π}{8} \sin^{2} (\frac{π}{2} - \frac{π}{8}) = \sin^{2} \frac{π}{8} \cos^{2} \frac{π}{8} = \frac{1}{4} {(2 \sin \frac{π}{8} \cos \frac{π}{8})}^{2} = \frac{1}{4} \sin^{2} \frac{π}{4} = \frac{1}{4} \times \frac{1}{2} = \frac{1}{8}$