sin^2-2sinxcosx+cos^2x<=0 inequation

Given the inequality:
$$\left(\sin^{2}{\left(x \right)} - 2 \sin{\left(x \right)} \cos{\left(x \right)}\right) + \cos^{2}{\left(x \right)} \leq 0$$
To solve this inequality, we must first solve the corresponding equation:
$$\left(\sin^{2}{\left(x \right)} - 2 \sin{\left(x \right)} \cos{\left(x \right)}\right) + \cos^{2}{\left(x \right)} = 0$$
Solve:
$$x_{1} = - \frac{3 \pi}{4}$$
$$x_{2} = \frac{\pi}{4}$$
$$x_{1} = - \frac{3 \pi}{4}$$
$$x_{2} = \frac{\pi}{4}$$
This roots
$$x_{1} = - \frac{3 \pi}{4}$$
$$x_{2} = \frac{\pi}{4}$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{3 \pi}{4} - \frac{1}{10}$$
=
$$- \frac{3 \pi}{4} - \frac{1}{10}$$
substitute to the expression
$$\left(\sin^{2}{\left(x \right)} - 2 \sin{\left(x \right)} \cos{\left(x \right)}\right) + \cos^{2}{\left(x \right)} \leq 0$$
$$\left(- 2 \sin{\left(- \frac{3 \pi}{4} - \frac{1}{10} \right)} \cos{\left(- \frac{3 \pi}{4} - \frac{1}{10} \right)} + \sin^{2}{\left(- \frac{3 \pi}{4} - \frac{1}{10} \right)}\right) + \cos^{2}{\left(- \frac{3 \pi}{4} - \frac{1}{10} \right)} \leq 0$$

   2/1    pi\      2/1    pi\        /1    pi\    /1    pi\     
cos |-- + --| + sin |-- + --| - 2*cos|-- + --|*sin|-- + --| <= 0
    \10   4 /       \10   4 /        \10   4 /    \10   4 /     

but

   2/1    pi\      2/1    pi\        /1    pi\    /1    pi\     
cos |-- + --| + sin |-- + --| - 2*cos|-- + --|*sin|-- + --| >= 0
    \10   4 /       \10   4 /        \10   4 /    \10   4 /     

Then
$$x \leq - \frac{3 \pi}{4}$$
no execute
one of the solutions of our inequality is:
$$x \geq - \frac{3 \pi}{4} \wedge x \leq \frac{\pi}{4}$$

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