Solve `tan^(3) x+3 gt 3tan x+tan^(2) x`.

Answer

`x in (n pi+pi/3, n pi+pi/2) uu (npi-pi/3, n pi+pi/4), n in Z`

Explanation

`tan^(3)x-tan^(2)x+3-3 tan x gt 0`
`rArr tan^(2) x(tan x-1)-3 (tan x-1) gt 0`
`rArr (tan x -1) (tan^(2) x-3) gt 0`
`rArr (tan x-1) (tan x+sqrt(3))(tan x-sqrt(3)) gt 0`
`rArr (y-1) (y+sqrt(3)) (y-sqrt(3)) gt 0`, where `tan x=y`
Sign-scheme of above inequality is as follows :

`:. -sqrt(3) lt y lt 1 or y gt sqrt(3)`
`rArr -sqrt(3) lt tan x lt 1 or tan x gt sqrt(3)`
For `-pi//2 lt x lt pi//2`, we have `-pi//3 lt x lt pi//4 or pi//3 lt x lt pi//2`
`:.` General solution is
`x in (n pi +pi/3, npi +pi/2)uu (n pi - pi/3, n pi +pi/4)` where `n in Z`.