Question

The value of $\cos \frac{\pi }{10}\cos \frac{2\pi }{10}\cos \frac{4\pi }{10}\cos \frac{8\pi }{10}\cos \frac{16\pi }{10}$ is:

• A. $\frac{\sqrt{10+2\sqrt{5}}}{64}$
• B. $-\frac{\cos \left(\pi /10\right)}{16}$
• C. $\frac{\cos \left(\pi /10\right)}{16}$
• D. $\frac{\sqrt{10+2\sqrt{5}}}{16}$

Answer 1

The correct option is B $-\frac{\cos \left(\pi /10\right)}{16}$

$\cos \left(\frac{\pi }{10}\right)\cos \left(\frac{2\pi }{10}\right)\cos \left(\frac{4\pi }{10}\right)\cos \left(\frac{8\pi }{10}\right)\cos \left(\frac{16\pi }{10}\right)$

Multiplying numerators and denominator by $2\sin \left(\pi /10\right)$

$\frac{1}{2\sin \left(\pi /10\right)}\times 2\sin \left(\frac{\pi }{10}\right)\cos \left(\frac{\pi }{10}\right)\cos \left(\frac{2\pi }{10}\right)\cos \left(\frac{4\pi }{10}\right)\cos \left(\frac{8\pi }{10}\right)\cos \left(\frac{16\pi }{10}\right)$

$\frac{1}{2\sin \left(\pi /10\right)}\sin \left(\frac{2\pi }{10}\right)\cos \left(\frac{2\pi }{10}\right)\cos \left(\frac{4\pi }{10}\right)\cos \left(\frac{8\pi }{10}\right)\cos \left(\frac{16\pi }{10}\right)$

Multiplying numerators and denominator by $2$, we get

$=\frac{1}{4\sin \left(\pi /10\right)}\times 2\sin \left(\frac{2\pi }{10}\right)\cos \left(\frac{2\pi }{10}\right)\cos \left(\frac{4\pi }{10}\right)\cos \left(\frac{8\pi }{10}\right)\cos \left(\frac{16\pi }{10}\right)$

$=\frac{1}{4\sin \left(\pi /10\right)}\times \sin \left(\frac{4\pi }{10}\right)\cos \left(\frac{4\pi }{10}\right)\cos \left(\frac{8\pi }{10}\right)\cos \left(\frac{16\pi }{10}\right)$

$[As,2\sin \left(\frac{2\pi }{10}\right)\cos \left(\frac{2\pi }{10}\right)=\sin \left(\frac{4\pi }{10}\right)]$

$=\frac{1}{32\sin \left(\pi /10\right)}\sin \left(\frac{32\pi }{10}\right)$

$=\frac{\sin (3\pi +\frac{2\pi }{10})}{32\sin \left(\pi /10\right)}$

$=\frac{-\sin \left(2\pi /10\right)}{32\sin \left(\pi /10\right)}$ $[As,\sin (3\pi +\theta )=-\sin \theta ]$

$\frac{-2\sin \left(\pi /10\right)\cos \left(\pi /10\right)}{32\sin \left(\pi /10\right)}$ [As, $\sin \left(2\pi /10\right)=2\sin \left(\pi /10\right)\cos \left(\pi /10\right)$]

$=-\frac{-1}{16}\cos \left(\pi /10\right)$