tg^2x-4tgx+3>0 inequation

Given the inequality:
$$\tan^{2}{\left(x \right)} - 4 \tan{\left(x \right)} + 3 > 0$$
To solve this inequality, we must first solve the corresponding equation:
$$\tan^{2}{\left(x \right)} - 4 \tan{\left(x \right)} + 3 = 0$$
Solve:
Given the equation:
$$\tan^{2}{\left(x \right)} - 4 \tan{\left(x \right)} + 3 = 0$$
Transform
$$\tan^{2}{\left(x \right)} - 4 \tan{\left(x \right)} + 3 = 0$$
$$\left(\tan{\left(x \right)} - 3\right) \left(\tan{\left(x \right)} - 1\right) = 0$$
Consider each factor separately

Step

$$\tan{\left(x \right)} - 1 = 0$$
- this is the simplest trigonometric equation
Move $-1$ to right part of the equation
with the change of sign in $-1$
We get:
$$\tan{\left(x \right)} = 1$$
This equation is transformed to
$$x = \pi n + \operatorname{atan}{\left(1 \right)}$$
Or
$$x = \pi n + \frac{\pi}{4}$$
, where n - is a integer

Step

$$\tan{\left(x \right)} - 3 = 0$$
- this is the simplest trigonometric equation
Move $-3$ to right part of the equation
with the change of sign in $-3$
We get:
$$\tan{\left(x \right)} = 3$$
This equation is transformed to
$$x = \pi n + \operatorname{atan}{\left(3 \right)}$$
Or
$$x = \pi n + \operatorname{atan}{\left(3 \right)}$$
, where n - is a integer
The final answer:
$$x_{1} = \pi n + \frac{\pi}{4}$$
$$x_{2} = \pi n + \operatorname{atan}{\left(3 \right)}$$
$$x_{1} = \pi n + \frac{\pi}{4}$$
$$x_{2} = \pi n + \operatorname{atan}{\left(3 \right)}$$
$$x_{1} = \pi n + \frac{\pi}{4}$$
$$x_{2} = \pi n + \operatorname{atan}{\left(3 \right)}$$
This roots
$$x_{1} = \pi n + \frac{\pi}{4}$$
$$x_{2} = \pi n + \operatorname{atan}{\left(3 \right)}$$
is the points with change the sign of the inequality expression.
First define with the sign to the leftmost point:
$$x_{0} < x_{1}$$
For example, let's take the point
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$\left(\pi n + \frac{\pi}{4}\right) - \frac{1}{10}$$
=
$$\pi n - \frac{1}{10} + \frac{\pi}{4}$$
substitute to the expression
$$\tan^{2}{\left(x \right)} - 4 \tan{\left(x \right)} + 3 > 0$$
$$\tan^{2}{\left(\pi n - \frac{1}{10} + \frac{\pi}{4} \right)} - 4 \tan{\left(\pi n - \frac{1}{10} + \frac{\pi}{4} \right)} + 3 > 0$$

       2/1    pi\        /1    pi\    
3 + cot |-- + --| - 4*cot|-- + --| > 0
        \10   4 /        \10   4 /    

one of the solutions of our inequality is:
$$x < \pi n + \frac{\pi}{4}$$

 _____           _____          
      \         /
-------ο-------ο-------
       x_1      x_2

Other solutions will get with the changeover to the next point
etc.
The answer:
$$x < \pi n + \frac{\pi}{4}$$
$$x > \pi n + \operatorname{atan}{\left(3 \right)}$$