Question

Find the value of the expression: $\cos \frac{\pi }{5}\cos \frac{2\pi }{5}\cos \frac{4\pi }{5}\cos \frac{8\pi }{5}$.

Answer 1

$\cos \frac{\pi }{5}\cos \frac{2\pi }{5}\cos \frac{4\pi }{5}\cos \frac{8\pi }{5}=\frac{2\sin \frac{\pi }{5}\cos \frac{\pi }{5}\cos \frac{2\pi }{5}\cos \frac{4\pi }{5}\cos \frac{8\pi }{5}}{2\sin \frac{\pi }{5}}=\frac{2\sin \frac{2\pi }{5}\cos \frac{2\pi }{5}\cos \frac{4\pi }{5}\cos \frac{8\pi }{5}}{2^2\sin \frac{\pi }{5}}=\frac{2\sin \frac{4\pi }{5}\cos \frac{4\pi }{5}\cos \frac{8\pi }{5}}{4\times 2\sin \frac{\pi }{5}}=\frac{2\sin \frac{8\pi }{5}\cos \frac{8\pi }{5}}{8\times 2\sin \frac{\pi }{5}}=\frac{\sin \frac{16\pi }{5}}{16\sin \frac{\pi }{5}}=\frac{\sin (3\pi +\frac{\pi }{5})}{16\sin \frac{\pi }{5}}=\frac{1}{16}$.