# Solve this trigonometric system $\tan x+\tan y=2\sqrt3 \land \tan\frac{x}2+\tan\frac{y}2=\frac{2\sqrt3}3$

$$\tan x+\tan y=2\sqrt3 \land \tan\frac{x}2+\tan\frac{y}2=\frac{2\sqrt3}3$$

I need full solution please. I've tried different transformations, but couldn't get much near, I keep getting huge polynomials. It should be possible with standard transformations and algorithms for solving systems only.

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$\tan\,x=\dfrac{2\tan\frac{x}{2}}{1-\tan^2\frac{x}{2}}$ is extremely useful... –  Ｊ. Ｍ. Feb 8 '12 at 13:41
I tried working with that, too. I'm probably missing something obvious afterwards, though :( –  Lazar Ljubenović Feb 8 '12 at 13:47
Elimination will have you dealing with the quartic $(3v-\sqrt{3})^2 (3v^2-2\sqrt{3}v-9)=0$, which has a double root and two simple roots. –  Ｊ. Ｍ. Feb 8 '12 at 13:49
$3\sqrt3a^4-12a^3-4\sqrt3a^2+16a-3\sqrt3=0$ I checked on WA, it's got the same solutions as your factorized polynomial. How can I get your polynomial by simple transformations? –  Lazar Ljubenović Feb 8 '12 at 14:31
Lazar, my hint was supposed to be a comment... posted from my phone (!). If you want me to elaborate, I can, but I think @J.M. 's comment is more helpful. –  The Chaz 2.0 Feb 8 '12 at 14:31