cos (pie/8).cos (3 pie/8). cos (5 pie/8). cos (7 pie/8)

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Solution

$\cos (\frac{π}{8}) . \cos (\frac{3 π}{8}) . \cos (\frac{5 π}{8}) . \cos (\frac{7 π}{8}) = \frac{1}{2} [2 \cos (\frac{7 π}{8}) \cos (\frac{π}{8})] . \frac{1}{2} [2 \cos (\frac{5 π}{8}) \cos (\frac{3 π}{8})] = \frac{1}{2} [\cos (\frac{7 π}{8} + \frac{π}{8}) + \cos (\frac{7 π}{8} - \frac{π}{8})] . \frac{1}{2} [\cos (\frac{5 π}{8} + \frac{3 π}{8}) + \cos (\frac{5 π}{8} - \frac{3 π}{8})] = \frac{1}{2} [\cos π + \cos (\frac{3 π}{4})] . \frac{1}{2} [\cos π + \cos (\frac{π}{4})] = \frac{1}{2} [- 1 + \cos (π - \frac{π}{4})] . \frac{1}{2} [- 1 + \frac{1}{\sqrt{2}}] = \frac{1}{4} (- 1 - \cos \frac{π}{4}) (- 1 + \frac{1}{\sqrt{2}}) = \frac{1}{4} (- 1 - \frac{1}{\sqrt{2}}) (- 1 + \frac{1}{\sqrt{2}}) = - \frac{1}{4} (\frac{1}{\sqrt{2}} + 1) (\frac{1}{\sqrt{2}} - 1) = - \frac{1}{4} [\frac{1}{2} - 1] = \frac{1}{8}$

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