Question

If
$16cos\left(\frac{2\pi }{15}\right)cos\left(\frac{4\pi }{15}\right)cos\left(\frac{8\pi }{15}\right)cos\left(\frac{16\pi }{15}\right)=n$ , The value of $n$ is....

• A. 1
• B. 0
• C. 3
• D. 2

Answer 1

The correct option is A 1

Given,

$16\cos \left(\frac{2\pi }{15}\right)\cos \left(\frac{4\pi }{15}\right)\cos \left(\frac{8\pi }{15}\right)\cos \left(\frac{16\pi }{15}\right)=n$

$=16\left(\cos A\cos 2A\cos 2^{2}A\cos 2^{3}A\right)$

where $A=\frac{2\pi }{15}$

$=16\left(\frac{\sin 2^{4}A}{2^4\sin A}\right)$

$=\frac{\sin 2^4\frac{2\pi }{15}}{\sin \frac{2\pi }{15}}$

$=\frac{\sin \frac{32\pi }{15}}{\sin \frac{2\pi }{15}}$

$=\frac{\sin (2\pi +\frac{2\pi }{15})}{\sin \frac{2\pi }{15}}$

$=\frac{\sin \frac{2\pi }{15}}{\sin \frac{2\pi }{15}}$

$=1$