Question

If P = cos $\frac{\pi }{20}.cos\frac{3\pi }{20}.cos\frac{7\pi }{20}.cos\frac{9\pi }{20}$ & Q = cos$\frac{\pi }{11}.cos\frac{2\pi }{11}.cos\frac{4\pi }{11}.cos\frac{8\pi }{11}.cos\frac{16\pi }{11},then\frac{P}{Q}$ is

• A. $notdefined$
• B. $1$
• C. $2$
• D. $noneofthese$

Answer 1

Answer: (C)

$2$

LHS let's $\frac{\pi }{20}=\theta$

P $=\cos \theta .\cos 3\theta .\cos 7\theta .\cos 9\theta$

( taking two term at a time i.e. $1^{st}$ & $4^{st}$ and $2^{st}$ & $3^{st}$)

$=2\cos \theta .\cos 9\theta \times \frac{1}{2}.2\cos 3\theta .\cos 7\theta \times \frac{1}{2}$

$=(\cos 10\theta +\cos 8\theta )\times (\cos 10\theta +\cos 4\theta )\times \frac{1}{4}$

since $\cos 10\theta =\cos \frac{10\pi }{20}=\cos \frac{\pi }{2}=0$

$=\cos 8\theta \times \cos 4\theta$

$=\cos 8\left(\frac{\pi }{20}\right)\times \cos 4\left(\frac{\pi }{20}\right)\times \frac{1}{4}$

$=\cos \frac{2\pi }{5}\times \cos \frac{\pi }{5}\times \frac{1}{4}$

$=\left(\frac{5^{.5}-1}{4}\right)\left(\frac{5.5+1}{4}\right)\times \frac{1}{4}$

$P$ $=\frac{1}{16}$

RHS let's $\frac{\pi }{11}=\Phi$

$=\cos \Phi .\cos 2\Phi .\cos 4\Phi \cos 8\Phi \cos 16\Phi$

$=\frac{\sin \frac{32\pi }{11}}{2^5\times \sin \frac{32\pi }{11}}$

$\text{Q}$ $=\frac{1}{32}$

Now, $\frac{P}{Q}=\frac{\frac{1}{16}}{\frac{1}{32}}$

$=2$