Question

Let $s_n = \cos \left(\frac{n\pi}{10}\right), n=1,2,3,\ldots$ Then the value of $\frac{s_{1}s_{2}\ldots s_{10}}{s_{1}+s_{2}+\ldots+s_{10}}$ is equal to

• A. $\frac{1}{\sqrt{2}}$
• B. $\frac{\sqrt{3}}{2}$
• C. $2\sqrt{2}$
• D. 0

Answer 1

Answer: (D)

0

Answer 2

Given, $s_{n}=\cos \left(\frac{n \pi}{10}\right)$
Now, $S_{1} \,S_{2}\, S_{3} \ldots S_{10}$
$=\cos \left(\frac{\pi}{10}\right) \cos \left(\frac{2 \pi}{10}\right) \ldots \cos \left(\frac{5 \pi}{10}\right) \ldots \cos \left(\frac{10 \pi}{10}\right)$
$=\cos \left(\frac{\pi}{10}\right) \cos \left(\frac{\pi}{5}\right) \ldots \cos \left(\frac{\pi}{2}\right) \ldots \cos (\pi)$
$=\cos \left(\frac{\pi}{10}\right) \cos \left(\frac{\pi}{5}\right) \ldots 0 \ldots \cos \pi=0$
$\therefore \frac{S_{1}\, S_{2}\, S_{3} \ldots S_{10}}{S_{1}+s_{2}+\ldots+s_{10}}=0$