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The value of $\tan \frac{\pi}{5}+2 \tan \frac{2 \pi}{5}+4 \cot \frac{4 \pi}{5}$ is
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$\cot \frac{\pi}{5}$
$\tan \frac{\pi}{5}+2 \tan \frac{2 \pi}{5}+4 \cot \frac{4 \pi}{5}$
$=\frac{\sin \frac{\pi}{5}}{\cos \frac{\pi}{5}}+2 \frac{\sin \frac{2 \pi}{5}}{\cos \frac{2 \pi}{5}}+4 \frac{\cos \frac{4 \pi}{5}}{\sin \frac{4 \pi}{5}}$
$=\frac{\sin \frac{\pi}{5}}{\cos \frac{\pi}{5}}+2 \frac{\sin \frac{2 \pi}{5}}{\cos \frac{2 \pi}{5}}+\frac{4\left(\cos ^{2} \frac{2 \pi}{5}-\sin ^{2} \frac{2 \pi}{5}\right)}{2 \sin \frac{2 \pi}{5} \cos \frac{2 \pi}{5}}$
$=\frac{\sin \frac{\pi}{5}}{\cos \frac{\pi}{5}}+2 \frac{\sin \frac{2 \pi}{5}}{\cos \frac{2 \pi}{5}}+\frac{2\left(\cos ^{2} \frac{2 \pi}{5}-\sin ^{2} \frac{2 \pi}{5}\right)}{\sin \frac{2 \pi}{5} \cos \frac{2 \pi}{5}}$
$=\frac{\sin \frac{\pi}{5}}{\cos \frac{\pi}{5}}+2\left[\frac{\sin ^{2} \frac{2 \pi}{5}+\cos ^{2} \frac{2 \pi}{5}-\sin ^{2} \frac{2 \pi}{5}}{\cos \frac{2 \pi}{5} \sin \frac{2 \pi}{5}}\right]$
$=\frac{\sin \frac{\pi}{5}}{\cos \frac{\pi}{5}}+\frac{2 \cos \frac{2 \pi}{5}}{\sin \frac{2 \pi}{5}}$
$=\frac{\sin \frac{\pi}{5}}{\cos \frac{\pi}{5}}+\frac{2\left(\cos ^{2} \frac{\pi}{5}-\sin ^{2} \frac{\pi}{5}\right)}{2 \sin \frac{\pi}{5} \cos \frac{\pi}{5}}$
$=\frac{\sin ^{2} \frac{\pi}{5}+\cos ^{2} \frac{\pi}{5}-\frac{\sin ^{2} \pi}{5}}{\sin \frac{\pi}{5} \cos \frac{\pi}{5}}$
$=\frac{\cos ^{2} \frac{\pi}{5}}{\sin \frac{\pi}{5} \cos \frac{\pi}{5}}$
$=\cot \frac{\pi}{5}$
$=\frac{\sin \frac{\pi}{5}}{\cos \frac{\pi}{5}}+2 \frac{\sin \frac{2 \pi}{5}}{\cos \frac{2 \pi}{5}}+4 \frac{\cos \frac{4 \pi}{5}}{\sin \frac{4 \pi}{5}}$
$=\frac{\sin \frac{\pi}{5}}{\cos \frac{\pi}{5}}+2 \frac{\sin \frac{2 \pi}{5}}{\cos \frac{2 \pi}{5}}+\frac{4\left(\cos ^{2} \frac{2 \pi}{5}-\sin ^{2} \frac{2 \pi}{5}\right)}{2 \sin \frac{2 \pi}{5} \cos \frac{2 \pi}{5}}$
$=\frac{\sin \frac{\pi}{5}}{\cos \frac{\pi}{5}}+2 \frac{\sin \frac{2 \pi}{5}}{\cos \frac{2 \pi}{5}}+\frac{2\left(\cos ^{2} \frac{2 \pi}{5}-\sin ^{2} \frac{2 \pi}{5}\right)}{\sin \frac{2 \pi}{5} \cos \frac{2 \pi}{5}}$
$=\frac{\sin \frac{\pi}{5}}{\cos \frac{\pi}{5}}+2\left[\frac{\sin ^{2} \frac{2 \pi}{5}+\cos ^{2} \frac{2 \pi}{5}-\sin ^{2} \frac{2 \pi}{5}}{\cos \frac{2 \pi}{5} \sin \frac{2 \pi}{5}}\right]$
$=\frac{\sin \frac{\pi}{5}}{\cos \frac{\pi}{5}}+\frac{2 \cos \frac{2 \pi}{5}}{\sin \frac{2 \pi}{5}}$
$=\frac{\sin \frac{\pi}{5}}{\cos \frac{\pi}{5}}+\frac{2\left(\cos ^{2} \frac{\pi}{5}-\sin ^{2} \frac{\pi}{5}\right)}{2 \sin \frac{\pi}{5} \cos \frac{\pi}{5}}$
$=\frac{\sin ^{2} \frac{\pi}{5}+\cos ^{2} \frac{\pi}{5}-\frac{\sin ^{2} \pi}{5}}{\sin \frac{\pi}{5} \cos \frac{\pi}{5}}$
$=\frac{\cos ^{2} \frac{\pi}{5}}{\sin \frac{\pi}{5} \cos \frac{\pi}{5}}$
$=\cot \frac{\pi}{5}$
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