# Solve the equation $\tan(2x) = 1+\tan(x)$

I am trying to solve the equation $$\tan(2x) = 1+\tan(x).$$

I have tried putting $u = tan(x),$ and $tan(2x) = \frac{2u}{1-u^2}$ so that $-u^3 + u^2 + 3u = 1,$ but I can't find any roots that would help me.

I have also tried using all the trigonometric identities I could think of but that hasn't helped me either, so I have a feeling that I should "see" something that I am failing to see. Any advice?

Edit: I should find all real solutions.

I think you may have just made an algebraic slip: $$\frac{2u}{1-u^2} = 1+u$$ implies $$2u = (1+u)(1-u^2)= 1 -u^3 - u^2 + u$$ and that $$u^3 + u^2 +u -1 = 0.$$
• For self-check, one (and the only) solution is $(\sqrt[3]{17 + \sqrt{297}} + \sqrt[3]{17-\sqrt{297}} - 1) / 3$ – Abstraction Aug 9 '16 at 8:33