P = cos (π/(15)).cos ((2π)/(15)).cos ((3π)/(15)).cos ((4π)/(15)).cos ((5π)/(15)).cos ((6π)/(15)).cos ((7π)/(15))


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Question Number 130469 by liberty last updated on 26/Jan/21

$P = \cos \frac{π}{15} . \cos \frac{2 π}{15} . \cos \frac{3 π}{15} . \cos \frac{4 π}{15} . \cos \frac{5 π}{15} . \cos \frac{6 π}{15} . \cos \frac{7 π}{15}$
	Answered by EDWIN88 last updated on 26/Jan/21

	$2 psin \frac{π}{15} = \sin \frac{2 π}{15} . \cos \frac{2 π}{15} . \cos \frac{3 π}{15} . \cos \frac{4 π}{15} . \cos \frac{π}{3} . \cos \frac{6 π}{15} . \cos \frac{7 π}{15}$ $4 psin \frac{π}{15} = \sin \frac{4 π}{15} . \cos \frac{3 π}{15} . \cos \frac{4 π}{15} . \frac{1}{2} . \cos \frac{6 π}{15} . \cos \frac{7 π}{15}$ $8 psin \frac{π}{15} = \frac{1}{2} . \sin \frac{8 π}{15} . \cos \frac{π}{5} . \cos \frac{2 π}{5} . \cos \frac{7 π}{15}$ $note : \sin x = \sin (π - x); \sin \frac{8 π}{15} = \sin \frac{7 π}{15}$ $8 p \sin \frac{π}{15} = \frac{1}{2} \sin \frac{7 π}{15} . \cos \frac{7 π}{15} . \cos \frac{π}{5} . \cos \frac{2 π}{5}$ $16 psin \frac{π}{15} = \frac{1}{2} \sin \frac{14 π}{15} . \cos \frac{π}{5} . \cos \frac{2 π}{5}$ $16 p \sin \frac{π}{15} = \frac{1}{2} \sin \frac{π}{15} . (\cos \frac{2 π}{5} . \cos \frac{π}{5})$ $16 p = \frac{1}{2} (\cos \frac{2 π}{5} . \cos \frac{π}{5})$ $note : \cos \frac{2 π}{5} = \cos 72 ° = \sin 18 ° = \frac{\sqrt{5} - 1}{4}$ $\cos \frac{π}{5} = \cos 36 ° = 1 - 2 \sin^{2} 18 °$ $= 1 - 2 (\frac{6 - 2 \sqrt{5}}{16}) = 1 - (\frac{3 - \sqrt{5}}{4})$ $= \frac{\sqrt{5} + 1}{4}$ $Thus p = \frac{1}{32} (\frac{\sqrt{5} - 1}{4}) (\frac{\sqrt{5} + 1}{4}) = \frac{1}{128}$
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