tan(x)+ tan(2x) = 1

tan(x) + tan(2x) = 1

tanx +   [2 tanx] / [1 - tan^2x]  = 1

[tan x - tan^3x + 2 tanx]  /  (1 - tan^2x]  = 1

tanx - tan^3x + 2 tanx  = 1 - tan^2x     simplify

tan^3x - tan^2x - 3tanx + 1   = 0        let tan x  = u   and we have

u^3 - u^2 - 3u + 1  = 0

And the solutions to this equation are :

u ≈ - 1.4812    →       tan-1(-1.4812) = u = about 124.02°  + n(180)°

u ≈ .31111     →       tan-1 (.31111) = u = about  17.28°  + n(180)°

u ≈ 2.1701   →        tan-1 (2.1701)  = u = about  65.26°  + n(180)°  

......    where n is some integer in each case............

Here's a graph which (hopefully) confirms our solutions......https://www.desmos.com/calculator/ekla1xindv