Question

The value of $\cos \frac{2\pi }{15}\cdot \cos \frac{4\pi }{15}\cdot \cos \frac{8\pi }{15}\cdot \cos \frac{16\pi }{15}$ is equal to

• A. $\frac{1}{16}$
• B. $\frac{1}{32}$
• C. $\frac{1}{64}$
• D. $\frac{1}{8}$

Answer 1

Answer: (A)

$\cos \frac{2\pi }{15}\cdot \cos \frac{4\pi }{15}\cos \frac{8\pi }{15}\cos \frac{16\pi }{15}$
$=\frac{1}{2\sin \frac{\pi }{15}}\cdot 2\sin \frac{2\pi }{15}\cdot \cos \frac{2\pi }{15}\cdot \cos \frac{4\pi }{15}\cdot \cos \frac{8\pi }{15}\cos \frac{16\pi }{15}$
$\Rightarrow \frac{1}{2\sin \frac{2\pi }{15}}\cdot \sin \frac{4\pi }{15}\cdot \cos \frac{4\pi }{15}\cdot \cos \frac{8\pi }{15}\cdot \cos \frac{16\pi }{15}$
$\Rightarrow \frac{1}{4\sin \frac{2\pi }{15}}\cdot \sin \frac{8\pi }{15}\cdot \cos \frac{8\pi }{15}\cdot \cos \frac{16\pi }{15}$
$=\frac{1}{8\sin \frac{2\pi }{15}}\cdot \sin \frac{16\pi }{15}\cdot \cos \frac{16\pi }{15}$
$=\frac{1}{16\sin \frac{2\pi }{15}}\cdot \sin \frac{32\pi }{15}$
$=\frac{1}{16\sin \frac{2\pi }{15}}\cdot \sin \left(2\pi +\frac{2\pi }{15}\right)$
$=\frac{1}{16\sin \frac{2\pi }{15}}\cdot \sin \frac{2\pi }{15}$
$=\frac{1}{16}$