<s>[INST] Given $\tan x+ \tan 2x=\frac{2}{\sqrt{3}}$, find $\tan x\cot 2x$ I can't solve this problem. I tried to find $\tan x$ directly by solving cubic equations but I failed.
The problem is to find $\tan x\cot 2x$ given that
$$\tan x+ \tan 2x=\frac{2}{\sqrt{3}}, \>\>\>\>\>0<x<\pi/4$$
How am I supposed to solve this problem?
 [/INST] If $t:=\tan x$ then $\frac{t(3-t^2)}{1-t^2}=\tfrac{2}{\sqrt{3}}$ so $t^3-\tfrac{2}{\sqrt{3}}t^2+3t+\tfrac{2}{\sqrt{3}}=0$. This has one real root, $t=\tfrac{2+33/a-a}{3\sqrt{3}}$ with $a:=\sqrt[3]{154+9\sqrt{443}}$. Now just calculate $\frac{1-t^2}{2}$.
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