cos (((2π)/(21)))cos (((4π)/(21)))cos (((8π)/(21)))cos (((10π)/(22)))cos (((16π)/(21)))cos (((20π)/(21)))=?


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Question Number 208252 by efronzo1 last updated on 09/Jun/24

$\cos (\frac{2 π}{21}) \cos (\frac{4 π}{21}) \cos (\frac{8 π}{21}) \cos (\frac{10 π}{22}) \cos (\frac{16 π}{21}) \cos (\frac{20 π}{21}) = ?$
	Answered by som(math1967) last updated on 09/Jun/24

	$l e t \frac{π}{21} = x$ $c o s 2 x c o s 4 x c o s 8 x c o s 10 x c o s 16 x c o s 20 x$ $c o s x c o s 2 x c o s 4 x c o s 8 x c o s 10 x c o s 5 x$ $[21 x = π \Rightarrow c o s 20 x = - c o s x, c o s 16 x = - c o s 5 x]$ $\frac{1}{4 s i n x s i n 5 x} \times s i n 2 x c o s 2 x c o s 4 x c o s 8 x s i n 10 x c o s 10 x$ $= \frac{1}{64 s i n x s i n 5 x} \times 2 s i n 8 x c o s 8 x \times 2 s i n 10 x c o s s 10 x$ $= \frac{1}{64 s i n x s i n 5 x} \times s i n 16 x s i n 20 x$ $= \frac{1}{64 s i n x s i n 5 x} \times s i n 5 x \times s i n x$ $[21 x = π \Rightarrow s i n 16 x = s i n 5 x, s i n 20 x = s i n x]$ $= \frac{1}{64}$
	Commented by efronzo1 last updated on 09/Jun/24


	Commented by som(math1967) last updated on 09/Jun/24

	$y e s$
	Commented by Ghisom last updated on 09/Jun/24

	$I get the same result$
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