<s>[INST] How to find all solutions of $\tan(x) = 2 + \tan(3x)$ without a calculator? Find all solutions of the equation $\tan(x) = 2 + \tan(3x)$ where $0<x<2\pi$.
By replacing $\tan(3x)$ with $\dfrac{\tan(2x) + \tan(x)}{1-\tan(2x)\tan(x)}$ 
I've gotten to $\tan^3 (x) - 3 \tan^2 (x) + \tan(x) + 1 =0$.
I am not sure how to proceed from there without the use of a calculator.
 [/INST] Put $t=\tan(x)$, so that your equation become:
$t^3-3t^2+t+1=0$.
Since $t^3-3t^2+t+1=(t-1)(t^2-2t-1)$, it follows that the solutions are $t=1$, $t=1\pm\sqrt 2$, from which
$x\equiv\arctan(1)\equiv\frac\pi 4\pmod \pi$,
$x\equiv\arctan(1+\sqrt 2)\equiv\frac{3\pi}8\pmod\pi$,
$x\equiv\arctan(1-\sqrt 2)\equiv-\frac\pi8\pmod\pi$.
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