prove that cos (π/(15)) cos ((2π)/(15)) cos ((3π)/(15)) cos ((4π)/(15)) cos ((5π)/(15)) cos ((6π)/(15)) cos ((7π)/(15)) = (1/2^7 )


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Question Number 37864 by kunal1234523 last updated on 18/Jun/18

$prove that$ $\cos \frac{π}{15} \cos \frac{2 π}{15} \cos \frac{3 π}{15} \cos \frac{4 π}{15} \cos \frac{5 π}{15} \cos \frac{6 π}{15} \cos \frac{7 π}{15} = \frac{1}{2^{7}}$
	Answered by tanmay.chaudhury50@gmail.com last updated on 18/Jun/18

	$c o s x . c o s 2 x . c o s 3 x . c o s 4 x . c o s 5 x . c o s 6 x . c o s 7 x = k$ $x = \frac{Π}{15}$ $2 s i n x . k = (2 s i n x c o s x) . c o s 2 x . c o z 3 x . c o z 4 x . c o s 5 x$ $c o s 6 x . c o s 7 x$ $2^{2} s i n x . k = (2 s i n 2 x c o s 2 x) c o s 3 x c o s 4 x c o s 5 x c o s 6 x$ $c o s 7 x$ $2^{3} s i n x . k = (2 s i n 4 x c o s 4 x) c o s 3 x c o s 5 x c o s 6 x c o 7 x$ $2^{3} s i n x . k = (s i n 8 x) c o s 3 x c o s 5 x c o z 6 x c o s 7 x$ $n o w 15 x = 7 x + 8 x$ $7 x + 8 x = Π$ $8 x = Π - 7 x$ $s i n (8 x) = s i n (Π - 7 x)$ $s i n 8 x = s i n 7 x$ $2^{4} s i n x . k = (2 s i n 7 x c o s 7 x) c o s 3 x c o s 5 x c o s 6 x$ $2^{4} s i n x . k = s i n 14 x . c o s (3 \times \frac{Π}{15}) (c o s 5 \times \frac{Π}{15}) c o s (6 \times \frac{Π}{15}$ $15 x = 14 x + x$ $Π = 14 x + x$ $14 x = Π - x$ $s i n (14 x) = s i n (Π - x) = s i n x$ $2^{4} . k = c o s 36^{o} \times c o s 60^{o} \times c o s 72^{o}$ $2^{4} k = c o s 36^{o} \times \frac{1}{2} \times s i n 18^{o}$ $2^{5} k = s i n 18^{o} \times c o s 36^{o}$ $2^{5} k = \frac{\sqrt{5} - 1}{4} \times \frac{\sqrt{5} + 1}{4}$ $2^{5} k = \frac{4}{16}$ $2^{5} \times 2^{2} \times k = 1$ $k = \frac{1}{2^{7}}$
	Answered by MrW3 last updated on 18/Jun/18

	$P = \cos \frac{π}{15} \cos \frac{2 π}{15} \cos \frac{3 π}{15} \cos \frac{4 π}{15} \cos \frac{5 π}{15} \cos \frac{6 π}{15} \cos \frac{7 π}{15}$ $2 \sin \frac{π}{15} P = 2 \sin \frac{π}{15} \cos \frac{π}{15} \cos \frac{2 π}{15} \cos \frac{3 π}{15} \cos \frac{4 π}{15} \cos \frac{5 π}{15} \cos \frac{6 π}{15} \cos \frac{7 π}{15}$ $2 \sin \frac{π}{15} P = \sin \frac{2 π}{15} \cos \frac{2 π}{15} \cos \frac{3 π}{15} \cos \frac{4 π}{15} \cos \frac{5 π}{15} \cos \frac{6 π}{15} \cos \frac{7 π}{15}$ $2^{2} \sin \frac{π}{15} P = 2 \sin \frac{2 π}{15} \cos \frac{2 π}{15} \cos \frac{3 π}{15} \cos \frac{4 π}{15} \cos \frac{5 π}{15} \cos \frac{6 π}{15} \cos \frac{7 π}{15}$ $2^{2} \sin \frac{π}{15} P = \sin \frac{4 π}{15} \cos \frac{3 π}{15} \cos \frac{4 π}{15} \cos \frac{5 π}{15} \cos \frac{6 π}{15} \cos \frac{7 π}{15}$ $2^{3} \sin \frac{π}{15} P = 2 \sin \frac{4 π}{15} \cos \frac{3 π}{15} \cos \frac{4 π}{15} \cos \frac{5 π}{15} \cos \frac{6 π}{15} \cos \frac{7 π}{15}$ $2^{3} \sin \frac{π}{15} P = \sin \frac{8 π}{15} \cos \frac{3 π}{15} \cos \frac{5 π}{15} \cos \frac{6 π}{15} \cos \frac{7 π}{15}$ $2^{4} \sin \frac{π}{15} P = 2 \sin \frac{8 π}{15} \cos \frac{3 π}{15} \cos \frac{5 π}{15} \cos \frac{6 π}{15} \cos \frac{7 π}{15}$ $2^{4} \sin \frac{π}{15} P = 2 \sin \frac{7 π}{15} \cos \frac{3 π}{15} \cos \frac{5 π}{15} \cos \frac{6 π}{15} \cos \frac{7 π}{15}$ $2^{4} \sin \frac{π}{15} P = \sin \frac{14 π}{15} \cos \frac{3 π}{15} \cos \frac{5 π}{15} \cos \frac{6 π}{15}$ $2^{4} \sin \frac{π}{15} P = \sin \frac{π}{15} \cos \frac{3 π}{15} \cos \frac{5 π}{15} \cos \frac{6 π}{15}$ $2^{4} P = \cos \frac{3 π}{15} \cos \frac{5 π}{15} \cos \frac{6 π}{15}$ $2^{5} \sin \frac{3 π}{15} P = 2 \sin \frac{3 π}{15} \cos \frac{3 π}{15} \cos \frac{5 π}{15} \cos \frac{6 π}{15}$ $2^{5} \sin \frac{3 π}{15} P = \sin \frac{6 π}{15} \cos \frac{5 π}{15} \cos \frac{6 π}{15}$ $2^{6} \sin \frac{3 π}{15} P = 2 \sin \frac{6 π}{15} \cos \frac{5 π}{15} \cos \frac{6 π}{15}$ $2^{6} \sin \frac{3 π}{15} P = \sin \frac{12 π}{15} \cos \frac{5 π}{15}$ $2^{6} \sin \frac{3 π}{15} P = \sin \frac{3 π}{15} \cos \frac{5 π}{15}$ $2^{6} P = \cos \frac{5 π}{15}$ $2^{6} P = \cos \frac{π}{3}$ $2^{6} P = \frac{1}{2}$ $\Rightarrow P = \frac{1}{2^{7}}$
	Commented by kunal1234523 last updated on 18/Jun/18

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