cos(x)<=2\2 inequation

Given the inequality:
$$\cos{\left(x \right)} \leq 1$$
To solve this inequality, we must first solve the corresponding equation:
$$\cos{\left(x \right)} = 1$$
Solve:
Given the equation
$$\cos{\left(x \right)} = 1$$
- this is the simplest trigonometric equation
This equation is transformed to
$$x = \pi n + \operatorname{acos}{\left(1 \right)}$$
$$x = \pi n - \pi + \operatorname{acos}{\left(1 \right)}$$
Or
$$x = \pi n$$
$$x = \pi n - \pi$$
, where n - is a integer
$$x_{1} = \pi n$$
$$x_{2} = \pi n - \pi$$
$$x_{1} = \pi n$$
$$x_{2} = \pi n - \pi$$
This roots
$$x_{1} = \pi n$$
$$x_{2} = \pi n - \pi$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$\pi n + - \frac{1}{10}$$
=
$$\pi n - \frac{1}{10}$$
substitute to the expression
$$\cos{\left(x \right)} \leq 1$$
$$\cos{\left(\pi n - \frac{1}{10} \right)} \leq 1$$

cos(-1/10 + pi*n) <= 1

one of the solutions of our inequality is:
$$x \leq \pi n$$

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

Other solutions will get with the changeover to the next point
etc.
The answer:
$$x \leq \pi n$$
$$x \geq \pi n - \pi$$