ctg5x>1 inequation

Given the inequality:
$$\cot{\left(5 x \right)} > 1$$
To solve this inequality, we must first solve the corresponding equation:
$$\cot{\left(5 x \right)} = 1$$
Solve:
Given the equation
$$\cot{\left(5 x \right)} = 1$$
transform
$$\cot{\left(5 x \right)} - 1 = 0$$
$$\cot{\left(5 x \right)} - 1 = 0$$
Do replacement
$$w = \cot{\left(5 x \right)}$$
Move free summands (without w)
from left part to right part, we given:
$$w = 1$$
We get the answer: w = 1
do backward replacement
$$\cot{\left(5 x \right)} = w$$
substitute w:
$$x_{1} = \frac{\pi}{20}$$
$$x_{1} = \frac{\pi}{20}$$
This roots
$$x_{1} = \frac{\pi}{20}$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{1}{10} + \frac{\pi}{20}$$
=
$$- \frac{1}{10} + \frac{\pi}{20}$$
substitute to the expression
$$\cot{\left(5 x \right)} > 1$$
$$\cot{\left(5 \left(- \frac{1}{10} + \frac{\pi}{20}\right) \right)} > 1$$

   /1   pi\    
tan|- + --| > 1
   \2   4 /    

the solution of our inequality is:
$$x < \frac{\pi}{20}$$

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