Question

solve: tanx+tan2x+√3tanx tan2x=√3 (0<=x<=360)

Answer 1

tanx+tan2x+√3tanx tan2x=√3
tanx+tan2x = √3 - √3tanxtan2x
tanx+tan2x = √3(1-tanx tan2x)
(tanx+tan2x)/(1 - tanx tan2x) = √3
tan3x = √3
...

Answer 2

tanx - ?3 + tan2x + ?3tanx tan2x = 0
tanx - ?3 + tan 2x(1 + ?3tanx) = 0

recall tan 2x = 2tanx/(1 - tan^2 x)

tanx - ?3 + 2tanx/(1 - tan^2 x)(1 + ?3tanx) = 0
times (1 - tan^2 x)


tanx - tan^3 x - ?3 + ?3tan^2 x + 2tanx(1 + ?3tanx) = 0
let tanx = y for easier typing
y - y^3 - ?3 - ?3y^2 + 2y(1 + ?3y) = 0
y - y^3 - ?3 - ?3y^2 + 2y + 2?3y^2 = 0
y^3 - ?3y^2 - 3y - ?3 = 0

not easy to solve a cubic, so I sent it to
Wolfram:
http://www.wolframalpha.com/input/?i=y+-+y%5E3+-+%E2%88%9A3+-+(%E2%88%9A3)y%5E2+%2B+2y+%2B+(2%E2%88%9A3)y%5E2+%3D+0

See if that helps

Answer 3

What a splendid observation !

Ignore my gibberish, it must contain an error somewhere anyhow.