Question

If the value of $cos\frac{2\pi }{15}\cdot cos\frac{4\pi }{15}\cdot cos\frac{8\pi }{15}\cdot cos\frac{14\pi }{15}$ is $\frac{1}{k}$ then find $k.$

Answer 1

Answer: 16

Given $cos\frac{2\pi }{15}\cdot cos\frac{4\pi }{15}\cdot cos\frac{8\pi }{15}\cdot cos\frac{14\pi }{15}$
$=cos\frac{2\pi }{15}\cdot cos\frac{4\pi }{15}\cdot cos\frac{8\pi }{15}\cdot cos\left(\pi -\frac{\pi }{15}\right)$
$=-cos\frac{\pi }{15}cos\frac{2\pi }{15}cos\frac{4\pi }{15}cos\frac{8\pi }{15}$
$=-\frac{sin\frac{16\pi }{15}}{2^{4}sin\frac{\pi }{15}}$
$=-\frac{sin\left(\pi +\frac{\pi }{15}\right)}{2^{4}sin\frac{\pi }{15}}=\frac{sin\left(\frac{\pi }{15}\right)}{2^{4}sin\left(\frac{\pi }{15}\right)}$
$=\frac{1}{16}$
$\Rightarrow k=16.$