$Tan\frac{\pi }{5}+2Tan\frac{2\pi }{5}+4\cot \frac{4\pi }{5}=$

Formula: $\cot \theta -\tan \theta =2\cot 2\theta$

$\cot \theta -\tan \theta =2\cot 2\theta$

$\tan \frac{\pi }{5}+2\tan \frac{2\pi }{5}+4\cot \frac{4\pi }{5}=\cot \frac{\pi }{5}-2\cot \frac{2\pi }{5}+2\tan \frac{2\pi }{5}+4\cot \frac{4\pi }{5}$

$\cot \frac{\pi }{5}-2\left[\cot \frac{2\pi }{5}-\tan \frac{2\pi }{5}\right]+4\cot \frac{4\pi }{5}=\cot \frac{\pi }{5}-4\cot \frac{4\pi }{5}+4\cot \frac{4\pi }{5}$

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