The value of tan frac{pi}{5} + 2 tan frac{2 p | vedclass

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(A) We know that $\cot 	heta = \frac{1}{	an 	heta}$. Also,$\cot \frac{4 \pi}{5} = \cot (\pi - \frac{\pi}{5}) = -\cot \frac{\pi}{5}$.Using the ident

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$\cot \frac{\pi}{5}$

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(A) We know that $\cot 	heta = \frac{1}{	an 	heta}$. Also,$\cot \frac{4 \pi}{5} = \cot (\pi - \frac{\pi}{5}) = -\cot \frac{\pi}{5}$.Using the identity $	an 	heta + \cot 	heta = \frac{2}{\sin 2 	heta}$,we simplify the expression.Let $S = 	an \frac{\pi}{5} + 2 	an \frac{2 \pi}{5} + 4 \cot \frac{4 \pi}{5}$.Note that $4 \cot \frac{4 \pi}{5} = -4 \cot \frac{\pi}{5}$.So,$S = 	an \frac{\pi}{5} + 2 	an \frac{2 \pi}{5} - 4 \cot \frac{\pi}{5}$.Using $	an 	heta - \cot 	heta = -2 \cot 2 	heta$,we have $	an \frac{\pi}{5} - \cot \frac{\pi}{5} = -2 \cot \frac{2 \pi}{5}$.Thus,$S = -2 \cot \frac{2 \pi}{5} + 2 	an \frac{2 \pi}{5} - 3 \cot \frac{\pi}{5}$ (This approach is complex).Alternatively,using the identity $	an 	heta + 2 	an 2 	heta + 4 	an 4 	heta + 8 \cot 8 	heta = \cot 	heta$,we can evaluate the expression.For the given expression,it simplifies to $\cot \frac{\pi}{5}$.

The value of $\tan \frac{\pi}{5} + 2 \tan \frac{2 \pi}{5} + 4 \cot \frac{4 \pi}{5}$ is

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