$\cot \theta \cot 3 \theta \cot 5 \theta \cot 7 \theta \cot 9 \theta \cot 11 \theta \cot 13 \theta \cot 15 \theta \cot 17 \theta \cot 19 \theta$ এবং $\theta = \frac{\pi}{20}$

Solve:

দেওয়া আছে, $\theta=\frac{\pi}{20} \Rightarrow \frac{\pi}{2}=10 \theta$$\cot \theta \cot 3 \theta \cot 5 \theta \cot 7 \theta$

$\cot 9 \theta \cot 11 \theta \cot 13 \theta \cot 15 \theta \cot 17 \theta$

$\cot 19 \theta$

$=\cot \theta \cot 3 \theta \cot 5 \theta \cot 7 \theta \cot 9 \theta$

$\cot (10 \theta+\theta) \cot (10 \theta+3 \theta)$

$\cot (10 \theta+5 \theta) \cot (10 \theta+7 \theta)$

$\cot (10 \theta+9 \dot{\theta})$

$=\cot \theta \cot 3 \theta \cot 5 \theta \cot 7 \theta \cot 9 \theta$

$\cot \left(\frac{\pi}{2}+\theta\right) \cot \left(\frac{\pi}{2}+3 \theta\right) \cot \left(\frac{\pi}{2}+5 \theta\right)$

$\cot \left(\frac{\pi}{2}+7 \theta\right) \cot \left(\frac{\pi}{2}+9 \theta\right)$

$=\frac{1}{\tan \theta \tan 3 \theta \tan 5 \theta \tan 7 \theta \tan 9 \theta}(-\tan \theta)$

$(-\tan 3 \theta)(-\tan 5 \theta)(-\tan 7 \theta)(-\tan 9 \theta)$

$=-1$