First slide

Multiple and sub- multiple Angles

Question

$(1 + \cos (π / 8)) (1 + \cos (3 π / 8)) (1 + \cos (5 π / 8))$ $(1 + \cos (7 π / 8)) =$

Moderate

A

$1 / 2$
B

$\cos (π / 8)$
C

$1 / 8$
D

$(1 + \sqrt{2}) / 2 \sqrt{2}$

Solution

We can write $(1 + \cos (π / 8)) (1 + \cos (3 π / 8))$
$(1 - \cos (3 π / 8)) (1 - \cos (π / 8))$
$= (1 - \cos^{2} (π / 8)) (1 - \cos^{2} (3 π / 8))$
$= \sin^{2} π / 8 \sin^{2} 3 π / 8$
$= \sin^{2} π / 8 \cos^{2} π / 8 = (1 / 4) \sin^{2} π / 4) = 1 / 8$

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