Prove that `(tan 3 x)/(tan x)` never lies between `1/3 and 3.`

Explanation

Let
`y=(tan x)/(tan 3x)`
`=(tan x(1-3tan^(2)x))/(3tan x-tan^(3)x)`
`=(1-3tan^(2)x)/(3-tan^(2)x)`
or `3y-(tan^(2)x)y=1-3tan^(2)x`
or `(y-3)tan^(2)x=3y-1`
or `tan^(2)x=(3y-1)/(y-3)`
or `(3y-1)/(y-3)ge0` (LHS is a prefect square)

From the sign scheme `yle(1)/(3)` or `ygt3`
Thus, y never lies between `1//3` and 3.