Find the value of $9\tan{10^{\circ}}+2\tan{20^{\circ}}+4\tan{40^{\circ}}-\tan{80^{\circ}}$ 
Find the value of
  $$9\tan{10^{\circ}}+2\tan{20^{\circ}}+4\tan{40^{\circ}}-\tan{80^{\circ}}$$

It seem the answer is $0$, see http://www.wolframalpha.com/input/?i=9tan(10)%2B2tan(20)%2B4tan(40)-tan(80).

with the help of $$\tan \theta = \cot \theta-2\cot 2\theta$$
Replace $\theta \rightarrow 2\theta$ , we have $$2\tan 2\theta =2\cot 2\theta -4\cot(4\theta)$$
similarly $$4\tan(4\theta) = 4\cot(4\theta)-4\cot(8\theta)$$
So $$\tan \theta+2\tan 2\theta+2\tan 4\theta = -8\cot(8\theta)$$
put $\theta = 10^\circ$
$$\tan 10^\circ+2\tan 20^\circ+4\tan 40^\circ =\cot (10^\circ) -8\cot(80^\circ) = \tan(80^\circ)-8\tan(10^\circ)$$
$$9\tan 10^\circ+2\tan 20^\circ+4\tan 40^\circ -\tan (10^\circ) = 0$$
