Solve the inequality: $$\cot x<-1\quad\mbox{for }x\in\mathbb{R}$$
Robert solved the task in the following steps:
(1) He determined the points at which the function $y = \cot x$ is not defined and, thus, determined its domain: $$D=\mathbb{R}\backslash\left\{k\cdot\pi; k\in\mathbb{Z}\right\}$$
(2) Robert found the solution of the equation $\cot x=-1$, which is the set: $$K_1=\left\{\frac{3\pi}{4}+k\cdot\pi;k\in\mathbb{Z}\right\}$$
(3) He stated that the cotangent function is decreasing over its domain, so the value of $\cot x$ will be less than $-1$ when $x$ is greater than $\frac{3\pi}{4}$. Thus, he wrote: $$\cot x<-1\Leftrightarrow x > \frac{3\pi}{4}$$
(4) Finally, he excluded points that are not in the domain of $\cot x$ from the result obtained in step (3) and wrote down the solution: $$K=\left(\frac{3\pi}{4};+\infty\right)\backslash \bigcup_{k\in\mathbb{Z}}\left\{k\cdot\pi\right\}$$ The solution is not correct. In which step did Robert make a mistake?
The mistake is in step (1). The domain of the cotangent function is not specified correctly.
The mistake is in step (2). The period in the solution of the equation $\cot x=-1$ is determined incorrectly.
The mistake is in step (3). The function $\cot x$ is not decreasing over the entire domain.
The mistake is in step (4). Points in which the function is not defined do not need to be excluded from the interval obtained in step (3).
Let’s present the correct solution. The function $\cot x$ is decreasing only on open intervals bounded by two adjacent points where the cotangent is not defined, i.e., on the intervals: $$\ldots,(-\pi;0),\ (0;\pi),\ (\pi;2\pi),\ (2\pi;3\pi),\ (3\pi;4\pi),\ldots$$ Therefore, we can find the solution to the inequality $\cot x<-1$ on each interval $(0+k\cdot\pi;\pi+k\cdot\pi)$ as: $$\frac{3\pi}{4}+k\cdot\pi<x<\pi+k\cdot\pi,\ \mbox{ where } k\in\mathbb{Z},$$ The solution can be written as the union of all the corresponding subintervals: $$K=\bigcup_{k\in\mathbb{Z}}\left(\frac{3\pi}{4}+k\cdot\pi;\pi+k\cdot\pi\right)$$