Question

How do you solve `tan2x cotx - 3 = 0`?

Answer 1

Solve: tan 2x.cot x - 3 = 0

Explanation:

Call t = tan x, we get

`(2t)/(1 - t^2)(1/t) - 3 =` `(2t)/(t(1 - t^2)) - ((3t)(1 - t^2))/(t(1 - t^2))` =

`(-t + 3t^3)/(t(1 - t^2)) = (3t^2 - 1)/(1 - t^2) = 0`

Conditions: t different to 0, and different to `+- 1`

`(3t^2 - 1) = 0 --> t^2 = 1/3 -> t = tan x = +- sqrt3/3`

`t = tan x = (sqrt3)/3 -> x = pi/6`

`t = tan x = -sqrt3/3 --> x = (5pi)/6`

Check with `x = (5pi)/6`
`tan 2x = tan ((10pi)/6) = tan ((5pi)/3) = - tan (pi/3) = - sqrt3`
`cot x = cot ((5pi)/6) = - sqrt3`
`f(x) = (-sqrt3)(-sqrt3) - 3 = 0`. OK