Question

Let $s_n=\cos \left(\frac{n\pi }{10}\right),n=1,2,3,\dots$ Then the value of $\frac{s_1{s}_2\dots s_{10}}{s_1+s_2+\dots +s_{10}}$ is equal to

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

Answer 1

Answer: (D)

Given, $s_n=\cos \left(\frac{n\pi }{10}\right)$
Now, $S_1{S}_2{S}_3\dots S_{10}$
$=\cos \left(\frac{\pi }{10}\right)\cos \left(\frac{2\pi }{10}\right)\dots \cos \left(\frac{5\pi }{10}\right)\dots \cos \left(\frac{10\pi }{10}\right)$
$=\cos \left(\frac{\pi }{10}\right)\cos \left(\frac{\pi }{5}\right)\dots \cos \left(\frac{\pi }{2}\right)\dots \cos (\pi )$
$=\cos \left(\frac{\pi }{10}\right)\cos \left(\frac{\pi }{5}\right)\dots 0\dots \cos \pi =0$
$\therefore \frac{S_1{S}_2{S}_3\dots S_{10}}{S_1+s_2+\dots +s_{10}}=0$