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  4. If the value of cos(2 π /15)⋅ cos(4 π /15)⋅ cos(8 π /15)⋅ cos(14 π /15) is (1/k) then find k.

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Q. 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 $k.$

NTA AbhyasNTA Abhyas 2022

Solution:

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