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

Prove that (tan3x)/(tanx) never lies between 1/3a n d3 .

tan 5x tan3x tan2x=

Solve (tan 3x - tan 2x)/(1+tan 3x tan 2x)=1 .

Prove that tan(cot^(-1)x)=cot(tan^(-1)x)

Prove that tan(cot^(-1)x)=cot(tan^(-1)x)

Solve (tan3x-tan2x)/(1+tan3xtan2x )=1

Prove that tan (2 tan^(-1) x ) = 2 tan (tan^(-1) x + tan^(-1) x^(3)) .

Prove that (tan 3x.cos 4x + cos 3 x. tan 3 x + tan 3 x . cos 2 x )/(sin 4 x + sin 3 x + sin 2 x ) = 1

Given that for all real x, the expression (x^(2)-2x+4)/(x^(2)+2x+4) lies between (1)/(3) and 3 the values between which the expression (9tan^(2)x+6tanx+4)/(9tan^(2)x-6tanx+4) lies are

Show that tan 3x tan 2x tan x = tan 3x - tan 2x - tan x .