Value of \(\cot\left(\dfrac{\pi}{20}\right)\cot\left(\dfrac{3\pi}{20}\right)\cot\left(\dfrac{5\pi}{20}\right)\cot\left(\dfrac{7\pi}{20}\right)\cot\left(\dfrac{9\pi}{20}\right)\)
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Convert each angle to degrees: \(\dfrac{\pi}{20}=9^\circ\), \(\dfrac{3\pi}{20}=27^\circ\), \(\dfrac{5\pi}{20}=45^\circ\), \(\dfrac{7\pi}{20}=63^\circ\), \(\dfrac{9\pi}{20}=81^\circ\).
So the product is \(\cot 9^\circ\cot 27^\circ\cot 45^\circ\cot 63^\circ\cot 81^\circ\).
Note \(\cot 45^\circ=1\). Also, using \(\cot(90^\circ-\theta)=\tan\theta\): \(\cot 81^\circ=\cot(90^\circ-9^\circ)=\tan 9^\circ\) and \(\cot 63^\circ=\cot(90^\circ-27^\circ)=\tan 27^\circ\).
So the product becomes \((\cot 9^\circ\tan 9^\circ)\times(\cot 27^\circ\tan 27^\circ)\times 1=1\times1\times1=1\).
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