Question

$\cot \frac{\pi }{20}\cot \frac{3\pi }{20}\cot \frac{5\pi }{20}\cot \frac{7\pi }{20}\cot \frac{9\pi }{20}\cot \frac{15\pi }{20}=$

• A. $1$
• B. $-1$
• C. $\sqrt{3}$
• D. $-\sqrt{3}$

Answer 1

The correct option is B $-1$

Given,

$\cot \frac{\pi }{20}\cot \frac{3\pi }{20}\cot \frac{5\pi }{20}\cot \frac{7\pi }{20}\cot \frac{9\pi }{20}\cot \frac{15\pi }{20}$

$\Rightarrow \cot 9\times \cot 27\times \cot 45\times \cot 63\times \cot 81\times cot135$

$\Rightarrow \cot 9\times cot27\times \cot 45\times \cot (90-27)\times \cot (90-9)\times \cot (90+45)$

$\Rightarrow \cot 9\times \cot 27\times \cot 45\times \tan 27\times \tan 9\times (-\tan 45)$

$\Rightarrow \frac{1}{\tan 9}\times \frac{1}{\tan 27}\times \tan 27\times \tan 9\times \cot 45\times (-\tan 45)$

$\Rightarrow \cot 45\times (-\tan 45)$

$\Rightarrow 1\times (-1)=-1$