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  4. tan (π/5)+2 tan (2 π/5)+4 cot (4 π/5) is equal to

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Q. $\tan \frac{\pi}{5}+2 \tan \frac{2 \pi}{5}+4 \cot \frac{4 \pi}{5}$ is equal to

EAMCETEAMCET 2015

Solution:

Given that,
$\tan \frac{\pi}{5}+2 \tan \frac{2 \pi}{5}+4 \cot \frac{4 \pi}{5}$
$=\tan \frac{\pi}{5}+2\left[2 \cot \frac{4 \pi}{5}+\tan \frac{2 \pi}{5}\right]$
$=\tan \frac{\pi}{5}+2 \cot \frac{2 \pi}{5}$
${[\because 2 \cot 2 A+\tan A} ]$
$=\tan \frac{\pi}{5}+\cot \frac{\pi}{5}-\tan \frac{\pi}{5}=\cot \frac{\pi}{5}$