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Question: \(\cos\frac{\pi}{5}\cos\frac{2\pi}{5}\cos\frac{4\pi}{5}\cos\frac{8\pi}{5} =\)...

cos⁡π5cos⁡2π5cos⁡4π5cos⁡8π5=\cos\frac{\pi}{5}\cos\frac{2\pi}{5}\cos\frac{4\pi}{5}\cos\frac{8\pi}{5} =

A

1/16

B

0

C

– 1/8

D

–1/16

Answer

–1/16

Explanation

Solution

cos⁡π5cos⁡2π5cos⁡4π5cos⁡8π5=sin⁡24π524sin⁡π5=sin⁡16π516sin⁡π5=sin⁡(3π+π5)16sin⁡π5\cos\frac{\pi}{5}\cos\frac{2\pi}{5}\cos\frac{4\pi}{5}\cos\frac{8\pi}{5} = \frac{\sin\frac{2^{4}\pi}{5}}{2^{4}\sin\frac{\pi}{5}} = \frac{\sin\frac{16\pi}{5}}{16\sin\frac{\pi}{5}} = \frac{\sin\left( 3\pi + \frac{\pi}{5} \right)}{16\sin\frac{\pi}{5}}

=−sin⁡π516sin⁡π5=−116= \frac{- \sin\frac{\pi}{5}}{16\sin\frac{\pi}{5}} = - \frac{1}{16}.