tan(x)+cot(x)>=1/(sqrt(3))+sqrt(3) inequation

Given the inequality:
$$\tan{\left(x \right)} + \cot{\left(x \right)} \geq \frac{1}{\sqrt{3}} + \sqrt{3}$$
To solve this inequality, we must first solve the corresponding equation:
$$\tan{\left(x \right)} + \cot{\left(x \right)} = \frac{1}{\sqrt{3}} + \sqrt{3}$$
Solve:
Given the equation
$$\tan{\left(x \right)} + \cot{\left(x \right)} = \frac{1}{\sqrt{3}} + \sqrt{3}$$
transform
$$\tan{\left(x \right)} + \cot{\left(x \right)} - \frac{4 \sqrt{3}}{3} - 1 = 0$$
$$\tan{\left(x \right)} + \cot{\left(x \right)} - \sqrt{3} - 1 - \frac{\sqrt{3}}{3} = 0$$
Do replacement
$$w = \cot{\left(x \right)}$$
Given the equation:
$$w + \tan{\left(x \right)} - \sqrt{3} - 1 - \frac{\sqrt{3}}{3} = 0$$
transform:
$$w + \tan{\left(x \right)} - \frac{4 \sqrt{3}}{3} - 1 = 0$$
Expand brackets in the left part

-1 + w - 4*sqrt3/3 + tanx = 0

Move free summands (without w)
from left part to right part, we given:
$$w + \tan{\left(x \right)} - \frac{4 \sqrt{3}}{3} = 1$$
Divide both parts of the equation by (w - 4*sqrt(3)/3 + tan(x))/w

w = 1 / ((w - 4*sqrt(3)/3 + tan(x))/w)

We get the answer: w = 1 - tan(x) + 4*sqrt(3)/3
do backward replacement
$$\cot{\left(x \right)} = w$$
substitute w:
$$x_{1} = \frac{\pi}{6}$$
$$x_{2} = \frac{\pi}{3}$$
$$x_{1} = \frac{\pi}{6}$$
$$x_{2} = \frac{\pi}{3}$$
This roots
$$x_{1} = \frac{\pi}{6}$$
$$x_{2} = \frac{\pi}{3}$$
is the points with change the sign of the inequality expression.
First define with the sign to the leftmost point:
$$x_{0} \leq x_{1}$$
For example, let's take the point
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{1}{10} + \frac{\pi}{6}$$
=
$$- \frac{1}{10} + \frac{\pi}{6}$$
substitute to the expression
$$\tan{\left(x \right)} + \cot{\left(x \right)} \geq \frac{1}{\sqrt{3}} + \sqrt{3}$$
$$\tan{\left(- \frac{1}{10} + \frac{\pi}{6} \right)} + \cot{\left(- \frac{1}{10} + \frac{\pi}{6} \right)} \geq \frac{1}{\sqrt{3}} + \sqrt{3}$$

                                   ___
   /1    pi\      /1    pi\    4*\/ 3 
cot|-- + --| + tan|-- + --| >= -------
   \10   3 /      \10   3 /       3   
                               

one of the solutions of our inequality is:
$$x \leq \frac{\pi}{6}$$

 _____           _____          
      \         /
-------•-------•-------
       x1      x2

Other solutions will get with the changeover to the next point
etc.
The answer:
$$x \leq \frac{\pi}{6}$$
$$x \geq \frac{\pi}{3}$$