tan(x)≥1 inequation

Given the inequality:
$$\tan{\left(x \right)} \geq 1$$
To solve this inequality, we must first solve the corresponding equation:
$$\tan{\left(x \right)} = 1$$
Solve:
Given the equation
$$\tan{\left(x \right)} = 1$$
- this is the simplest trigonometric equation
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
$$x_{1} = \pi n + \frac{\pi}{4}$$
$$x_{1} = \pi n + \frac{\pi}{4}$$
This roots
$$x_{1} = \pi n + \frac{\pi}{4}$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$\left(\pi n + \frac{\pi}{4}\right) - \frac{1}{10}$$
=
$$\pi n - \frac{1}{10} + \frac{\pi}{4}$$
substitute to the expression
$$\tan{\left(x \right)} \geq 1$$
$$\tan{\left(\pi n - \frac{1}{10} + \frac{\pi}{4} \right)} \geq 1$$

   /1    pi\     
cot|-- + --| >= 1
   \10   4 /     

but

   /1    pi\    
cot|-- + --| < 1
   \10   4 /    

Then
$$x \leq \pi n + \frac{\pi}{4}$$
no execute
the solution of our inequality is:
$$x \geq \pi n + \frac{\pi}{4}$$

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