Question

Simplify: $\cos \frac{\pi }{15}\cos \frac{2\pi }{15}\cos \frac{3\pi }{15}\cos \frac{4\pi }{15}\cos \frac{5\pi }{13}\cos \frac{6\pi }{15}\cos \frac{7\pi }{15}$

Answer 1

$\cos \frac{\pi }{15}\cos \frac{2\pi }{15}\cos \frac{3\pi }{15}\cos \frac{4\pi }{15}\cos \frac{5\pi }{15}\cos \frac{6\pi }{15}\cos \frac{7\pi }{15}$

$=\cos \frac{\pi }{15}\cos \frac{2\pi }{15}(-\cos \frac{12\pi }{15})\cos \frac{4\pi }{15}\cos \frac{\pi }{3}\cos \frac{6\pi }{15}(-\cos \frac{8\pi }{15})$

$=\left(\cos \frac{\pi }{15}\cos \frac{2\pi }{15}\cos \frac{4\pi }{15}\cos \frac{8\pi }{15}\right)\left(\cos \frac{6\pi }{15}\cos \frac{12\pi }{15}\right)\cos \frac{\pi }{3}$

$=\left(\frac{2\sin \pi /15}{2\sin \pi /15}\cos \frac{\pi }{15}\cos \frac{2\pi }{15}\cos \frac{4\pi }{15}\cos \frac{8\pi }{15}\right)\left(\cos \frac{2\pi }{5}\cos \frac{4\pi }{5}\right)\cos \frac{\pi }{3}$

$=\left(\frac{\sin 2\pi /15}{2\sin \pi /15}\cos \frac{2\pi }{15}\cos \frac{4\pi }{15}\cos \frac{8\pi }{15}\right)\left(\frac{2\sin 2\pi /5}{2\sin 2\pi /5}\cos \frac{2\pi }{5}\cos \frac{4\pi }{5}\right)\frac{1}{2}$

$=\left(\frac{\sin 4\pi /15\cos 4\pi /15\cos 8\pi /15}{4\sin \pi /15}\right)\left(\frac{\sin 4\pi /5\cos 4\pi /5}{2\sin 2\pi /5}\right)\frac{1}{2}$

$=\left(\frac{\sin 8\pi /15\cos 8\pi /15}{8\sin \pi /15}\right)\left(\frac{\sin 8\pi /5}{4\sin 2\pi /5}\right)\frac{1}{2}$

$=\left(\frac{\sin 16\pi /15}{16\sin \pi /15}\right)\left(\frac{\sin 8\pi /5}{4\sin 2\pi /5}\right)\frac{1}{2}$

$=\left(\frac{-\sin \pi /15}{16\sin \pi /15}\right)\left(\frac{-\sin 2\pi /5}{4\sin 2\pi /5}\right)\frac{1}{2}$

$=(-\frac{1}{16})(-\frac{1}{4})\frac{1}{2}=\frac{1}{128}$