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  4. Let sn = cos ((nπ/10)), n=1,2,3, ldots Then the value of (s1s2 ldots s10/s1+s2+ ldots+s10) is equal to

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Q. 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

KEAMKEAM 2014Trigonometric Functions

Solution:

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