( 1 + cos 8 ) ( 1 + cos 3 8 ) ( 1 + cos 5 8 ) ( 1 + cos 7 8 ) is equal to

Question

$(1 + \cos \frac{π}{8}) (1 + \cos \frac{3 π}{8}) (1 + \cos \frac{5 π}{8}) (1 + \cos \frac{7 π}{8})$ is equal to

Smart Achievers · Accepted Answer

(3)

Given expression,

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

$= (1 - \cos^{2} \frac{π}{8}) (1 - \cos^{2} \frac{3 π}{8})$

$= \sin^{2} \frac{π}{8} \sin^{2} \frac{3 π}{8} = \sin^{2} \frac{π}{8} \cos^{2} \frac{π}{8}$

$= \frac{1}{4} \sin^{2} \frac{π}{4} = \frac{1}{8}$

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