Chứng minh cos pi/15*cos 2pi/15*...*cos 7pi/15=1/128

\(\cos\dfrac{\pi}{15}.\cos\dfrac{2\pi}{15}...\cos\dfrac{7\pi}{15}=-\dfrac{1}{2}.\left(\cos\dfrac{\pi}{15}.\cos\dfrac{2\pi}{15}.\cos\dfrac{4\pi}{15}.\cos\dfrac{8\pi}{15}\right).\left(\cos\dfrac{3\pi}{15}.\cos\dfrac{6\pi}{15}\right)\)

\(=-\dfrac{1}{2}.\left(\cos\dfrac{\pi}{15}.\cos\left(2.\dfrac{\pi}{15}\right).\cos\left(2^2.\dfrac{\pi}{15}\right).\cos\left(2^3\dfrac{\pi}{15}\right)\right).\left(\cos\dfrac{3\pi}{15}.\cos\left(2.\dfrac{3\pi}{15}\right)\right)\)

\(=-\dfrac{1}{2}.\left(\dfrac{\sin\left(2^4.\dfrac{\pi}{15}\right)}{16.\sin\left(\dfrac{\pi}{15}\right)}\right).\left(\dfrac{\sin\left(2^2\dfrac{3\pi}{15}\right)}{4.\sin\left(\dfrac{3\pi}{15}\right)}\right)\)

\(=-\dfrac{1}{2}.\left(\dfrac{\sin\left(\dfrac{16\pi}{15}\right)}{16.\sin\left(\dfrac{\pi}{15}\right)}\right).\left(\dfrac{\sin\left(\dfrac{12\pi}{15}\right)}{4.\sin\left(\dfrac{3\pi}{15}\right)}\right)\)

\(=-\dfrac{1}{2}.\left(\dfrac{-\sin\left(\dfrac{\pi}{15}\right)}{16.\sin\left(\dfrac{\pi}{15}\right)}\right).\left(\dfrac{\sin\left(\dfrac{3\pi}{15}\right)}{4.\sin\left(\dfrac{3\pi}{15}\right)}\right)=\dfrac{1}{128}\)

$\cos \frac{\pi }{15}.\cos \frac{2\pi }{15}...\cos \frac{7\pi }{15}=-\frac{1}{2}.\left(\cos \frac{\pi }{15}.\cos \frac{2\pi }{15}.\cos \frac{4\pi }{15}.\cos \frac{8\pi }{15}\right).\left(\cos \frac{3\pi }{15}.\cos \frac{6\pi }{15}\right)$

$=-\frac{1}{2}.\left(\cos \frac{\pi }{15}.\cos \left(2.\frac{\pi }{15}\right).\cos \left(2^2.\frac{\pi }{15}\right).\cos \left(2^3\frac{\pi }{15}\right)\right).\left(\cos \frac{3\pi }{15}.\cos \left(2.\frac{3\pi }{15}\right)\right)$

  bởi Dương Qúy Thành 12/05/2019

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