Question

Solve: $\cos \frac{\pi }{5}\cos \frac{2\pi }{5}\cos \frac{4\pi }{5}\cos \frac{8\pi }{5}$

Answer 1

$cos\frac{\pi }{5}cos\frac{2\pi }{5}cos\frac{4\pi }{5}cos\frac{8\pi }{5}$

$=\frac{1}{2sin\frac{\pi }{5}}\left[2sin\frac{\pi }{5}cos\frac{\pi }{5}cos\frac{2\pi }{5}cos\frac{4\pi }{5}cos\frac{8\pi }{5}\right]$

$=\frac{1}{2sin\frac{\pi }{5}}\left[2sin\frac{2\pi }{5}cos\frac{2\pi }{5}cos\frac{4\pi }{5}cos\frac{8\pi }{5}\right]$

$=\frac{1}{4sin\frac{\pi }{5}}\left[sin\frac{4\pi }{5}cos\frac{4\pi }{5}cos\frac{8\pi }{5}\right]$

$=\frac{1}{8sin\frac{\pi }{5}}\left[sin\frac{8\pi }{5}cos\frac{8\pi }{5}\right]$

$=\frac{1}{16sin\frac{\pi }{5}}\left[2sin\frac{8\pi }{5}cos\frac{8\pi }{5}\right]=\frac{sin\frac{16\pi }{5}}{16sin\frac{\pi }{5}}=\frac{sin(3\pi +\frac{\pi }{5})}{16sin\frac{\pi }{5}}$

$=\frac{-sin\frac{\pi }{5}}{16sin\frac{\pi }{5}}$

$=\frac{-1}{16}$