The value of cot frac{pi}{20} cot frac{3 pi}{ | vedclass

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Question

(D) Given expression: $\cot \frac{\pi}{20} \cdot \cot \frac{3 \pi}{20} \cdot \cot \frac{5 \pi}{20} \cdot \cot \frac{7 \pi}{20} \cdot \cot \frac{9 \pi}

Vedclass · Accepted Answer

$1$

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(D) Given expression: $\cot \frac{\pi}{20} \cdot \cot \frac{3 \pi}{20} \cdot \cot \frac{5 \pi}{20} \cdot \cot \frac{7 \pi}{20} \cdot \cot \frac{9 \pi}{20}$Since $\pi = 180^{\circ}$,we have:$= \cot 9^{\circ} \cdot \cot 27^{\circ} \cdot \cot 45^{\circ} \cdot \cot 63^{\circ} \cdot \cot 81^{\circ}$Using the identity $\cot(90^{\circ} - 	heta) = 	an 	heta$:$= \cot 9^{\circ} \cdot \cot 27^{\circ} \cdot \cot 45^{\circ} \cdot \cot(90^{\circ} - 27^{\circ}) \cdot \cot(90^{\circ} - 9^{\circ})$$= \cot 9^{\circ} \cdot \cot 27^{\circ} \cdot \cot 45^{\circ} \cdot 	an 27^{\circ} \cdot 	an 9^{\circ}$Since $	an 	heta \cdot \cot 	heta = 1$:$= (\cot 9^{\circ} \cdot 	an 9^{\circ}) \cdot (\cot 27^{\circ} \cdot 	an 27^{\circ}) \cdot \cot 45^{\circ}$$= 1 \cdot 1 \cdot 1 = 1$

The value of $\cot \frac{\pi}{20} \cot \frac{3 \pi}{20} \cot \frac{5 \pi}{20} \cot \frac{7 \pi}{20} \cot \frac{9 \pi}{20}$ is

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