Question

The value of $\cos \frac{2\pi }{15}\cos \frac{4\pi }{15}\cos \frac{8\pi }{15}\cos \frac{14\pi }{15}$ is

• A. $\frac{1}{4}$
• B. $\frac{1}{8}$
• C. $\frac{1}{16}$
• D. None of these

Answer 1

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

The given expression $=\cos \frac{2\pi }{15}\cos \frac{4\pi }{15}\cos \frac{8\pi }{15}\cos \frac{14\pi }{15}$

$[\because \cos \left(\frac{16\pi }{15}\right)=\cos (2\pi -\frac{14\pi }{15})=\cos \frac{14\pi }{15}]$

$=\cos A.\cos 2A.\cos 2^{2}A.\cos 2^{3}A$, where $A=\frac{2\pi }{15}$

$=\frac{\sin 2^{4}A}{2^4\sin A}=\frac{\sin \frac{32\pi }{15}}{16\sin \frac{2\pi }{15}}=\frac{\sin (2\pi +\frac{2\pi }{15})}{16\sin \frac{2\pi }{15}}=\frac{\sin \frac{2\pi }{15}}{16\sin \frac{2\pi }{15}}=\frac{1}{16}$