Question

How do you solve `Tan x + cot x -2 =0`?

Answer 1

The solution is `x=\pi/4+npi`.

Explanation:

`cot(x)=1/tan(x)` then

`tan(x)+cot(x)-2=0`

`tan(x)+1/tan(x)-2=0`

`(tan(x)^2+1-2tan(x))/tan(x)=0`

`tan(x)^2+1-2tan(x)=0`

this is a square

`tan(x)^2+1-2tan(x)=0`

`(tan(x)-1)^2=0`

doing the square root we obtain

`tan(x)-1=0`

`tan(x)=1`

`x=arctan(1)=\pi/4`.

This is called the principal solution. We know that the tangent is periodic of period `pi` then also `\pi/4+pi`, `\pi/4+2pi`, `\pi/4+3pi`, etc. are solutions.
The generic solution is then `pi/4+npi` where `n` is any positive or negative integer number.