Question

Find the value of $\cos \frac{2\pi }{15}\cos \frac{4\pi }{15}\cos \frac{8\pi }{15}\cos \frac{14\pi }{15}$

• A. $\frac{1}{16}$
• B. $\frac{1}{8}$
• C. $\frac{3}{4}$
• D. $\frac{1}{4}$

Answer 1

The correct option is A $\frac{1}{16}$

$\cos \frac{2\pi }{15}\cos \frac{4\pi }{15}\cos \frac{8\pi }{15}\cos \frac{14\pi }{15}$
$=\frac{2\sin \frac{2\pi }{15}}{2\sin \frac{2\pi }{15}}\cos \frac{2\pi }{15}\cos \frac{4\pi }{15}\cos \frac{8\pi }{15}\cos \frac{14\pi }{15}$
$=\frac{\sin \frac{4\pi }{15}\cos \frac{4\pi }{15}\cos \frac{8\pi }{15}\cos \frac{14\pi }{15}}{2\sin \frac{2\pi }{15}}$
$=\frac{2\sin \frac{8\pi }{15}\cos \frac{8\pi }{15}\cos \frac{14\pi }{15}}{8\sin \frac{2\pi }{15}}$
$=\frac{2\sin \frac{16\pi }{15}\cos \frac{14\pi }{15}}{16\sin \frac{2\pi }{15}}$
$=\frac{\sin 2\pi +\sin \frac{2\pi }{15}}{16\sin \frac{2\pi }{15}}$
$=\frac{1}{16}$