2*sin(x)^(2)-2sqrt3*sin(x)*cos(x)>sqrt2+1 desigualdades

Se da la desigualdad:
$$- 2 \sqrt{3} \sin{\left(x \right)} \cos{\left(x \right)} + 2 \sin^{2}{\left(x \right)} > 1 + \sqrt{2}$$
Para resolver esta desigualdad primero hay que resolver la ecuación correspondiente:
$$- 2 \sqrt{3} \sin{\left(x \right)} \cos{\left(x \right)} + 2 \sin^{2}{\left(x \right)} = 1 + \sqrt{2}$$
Resolvemos:
$$x_{1} = 2 \operatorname{atan}{\left(- \sqrt{3} + \sqrt{2} \sqrt{3 - 2 \sqrt{2}} + \sqrt{1 + \left(- \sqrt{3} + \sqrt{2} \sqrt{3 - 2 \sqrt{2}} + \sqrt{6}\right)^{2}} + \sqrt{6} \right)}$$
$$x_{2} = - 2 \operatorname{atan}{\left(- \sqrt{6} - \sqrt{2} \sqrt{3 - 2 \sqrt{2}} + \sqrt{1 + \left(- \sqrt{3} + \sqrt{2} \sqrt{3 - 2 \sqrt{2}} + \sqrt{6}\right)^{2}} + \sqrt{3} \right)}$$
$$x_{3} = - 2 \operatorname{atan}{\left(- \sqrt{6} + \sqrt{2} \sqrt{3 - 2 \sqrt{2}} + \sqrt{\left(- \sqrt{6} + \sqrt{2} \sqrt{3 - 2 \sqrt{2}} + \sqrt{3}\right)^{2} + 1} + \sqrt{3} \right)}$$
$$x_{4} = - 2 \operatorname{atan}{\left(- \sqrt{6} - \sqrt{\left(- \sqrt{6} + \sqrt{2} \sqrt{3 - 2 \sqrt{2}} + \sqrt{3}\right)^{2} + 1} + \sqrt{2} \sqrt{3 - 2 \sqrt{2}} + \sqrt{3} \right)}$$
$$x_{1} = 2 \operatorname{atan}{\left(- \sqrt{3} + \sqrt{2} \sqrt{3 - 2 \sqrt{2}} + \sqrt{1 + \left(- \sqrt{3} + \sqrt{2} \sqrt{3 - 2 \sqrt{2}} + \sqrt{6}\right)^{2}} + \sqrt{6} \right)}$$
$$x_{2} = - 2 \operatorname{atan}{\left(- \sqrt{6} - \sqrt{2} \sqrt{3 - 2 \sqrt{2}} + \sqrt{1 + \left(- \sqrt{3} + \sqrt{2} \sqrt{3 - 2 \sqrt{2}} + \sqrt{6}\right)^{2}} + \sqrt{3} \right)}$$
$$x_{3} = - 2 \operatorname{atan}{\left(- \sqrt{6} + \sqrt{2} \sqrt{3 - 2 \sqrt{2}} + \sqrt{\left(- \sqrt{6} + \sqrt{2} \sqrt{3 - 2 \sqrt{2}} + \sqrt{3}\right)^{2} + 1} + \sqrt{3} \right)}$$
$$x_{4} = - 2 \operatorname{atan}{\left(- \sqrt{6} - \sqrt{\left(- \sqrt{6} + \sqrt{2} \sqrt{3 - 2 \sqrt{2}} + \sqrt{3}\right)^{2} + 1} + \sqrt{2} \sqrt{3 - 2 \sqrt{2}} + \sqrt{3} \right)}$$
Las raíces dadas
$$x_{3} = - 2 \operatorname{atan}{\left(- \sqrt{6} + \sqrt{2} \sqrt{3 - 2 \sqrt{2}} + \sqrt{\left(- \sqrt{6} + \sqrt{2} \sqrt{3 - 2 \sqrt{2}} + \sqrt{3}\right)^{2} + 1} + \sqrt{3} \right)}$$
$$x_{2} = - 2 \operatorname{atan}{\left(- \sqrt{6} - \sqrt{2} \sqrt{3 - 2 \sqrt{2}} + \sqrt{1 + \left(- \sqrt{3} + \sqrt{2} \sqrt{3 - 2 \sqrt{2}} + \sqrt{6}\right)^{2}} + \sqrt{3} \right)}$$
$$x_{4} = - 2 \operatorname{atan}{\left(- \sqrt{6} - \sqrt{\left(- \sqrt{6} + \sqrt{2} \sqrt{3 - 2 \sqrt{2}} + \sqrt{3}\right)^{2} + 1} + \sqrt{2} \sqrt{3 - 2 \sqrt{2}} + \sqrt{3} \right)}$$
$$x_{1} = 2 \operatorname{atan}{\left(- \sqrt{3} + \sqrt{2} \sqrt{3 - 2 \sqrt{2}} + \sqrt{1 + \left(- \sqrt{3} + \sqrt{2} \sqrt{3 - 2 \sqrt{2}} + \sqrt{6}\right)^{2}} + \sqrt{6} \right)}$$
son puntos de cambio del signo de desigualdad en las soluciones.
Primero definámonos con el signo hasta el punto extremo izquierdo:
$$x_{0} < x_{3}$$
Consideremos, por ejemplo, el punto
$$x_{0} = x_{3} - \frac{1}{10}$$
=
$$- 2 \operatorname{atan}{\left(- \sqrt{6} + \sqrt{2} \sqrt{3 - 2 \sqrt{2}} + \sqrt{\left(- \sqrt{6} + \sqrt{2} \sqrt{3 - 2 \sqrt{2}} + \sqrt{3}\right)^{2} + 1} + \sqrt{3} \right)} - \frac{1}{10}$$
=
$$- 2 \operatorname{atan}{\left(- \sqrt{6} + \sqrt{2} \sqrt{3 - 2 \sqrt{2}} + \sqrt{\left(- \sqrt{6} + \sqrt{2} \sqrt{3 - 2 \sqrt{2}} + \sqrt{3}\right)^{2} + 1} + \sqrt{3} \right)} - \frac{1}{10}$$
lo sustituimos en la expresión
$$- 2 \sqrt{3} \sin{\left(x \right)} \cos{\left(x \right)} + 2 \sin^{2}{\left(x \right)} > 1 + \sqrt{2}$$
$$- 2 \sqrt{3} \sin{\left(- 2 \operatorname{atan}{\left(- \sqrt{6} + \sqrt{2} \sqrt{3 - 2 \sqrt{2}} + \sqrt{\left(- \sqrt{6} + \sqrt{2} \sqrt{3 - 2 \sqrt{2}} + \sqrt{3}\right)^{2} + 1} + \sqrt{3} \right)} - \frac{1}{10} \right)} \cos{\left(- 2 \operatorname{atan}{\left(- \sqrt{6} + \sqrt{2} \sqrt{3 - 2 \sqrt{2}} + \sqrt{\left(- \sqrt{6} + \sqrt{2} \sqrt{3 - 2 \sqrt{2}} + \sqrt{3}\right)^{2} + 1} + \sqrt{3} \right)} - \frac{1}{10} \right)} + 2 \sin^{2}{\left(- 2 \operatorname{atan}{\left(- \sqrt{6} + \sqrt{2} \sqrt{3 - 2 \sqrt{2}} + \sqrt{\left(- \sqrt{6} + \sqrt{2} \sqrt{3 - 2 \sqrt{2}} + \sqrt{3}\right)^{2} + 1} + \sqrt{3} \right)} - \frac{1}{10} \right)} > 1 + \sqrt{2}$$

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      |           |           /      /                         _____________\                      _____________||              |           |           /      /                         _____________\                      _____________||    |           |           /      /                         _____________\                      _____________||         ___
     2|1          |  ___     /       |  ___     ___     ___   /         ___ |       ___     ___   /         ___ ||       ___    |1          |  ___     /       |  ___     ___     ___   /         ___ |       ___     ___   /         ___ ||    |1          |  ___     /       |  ___     ___     ___   /         ___ |       ___     ___   /         ___ || > 1 + \/ 2 
2*sin |-- + 2*atan\\/ 3  + \/    1 + \\/ 3  - \/ 6  + \/ 2 *\/  3 - 2*\/ 2  /   - \/ 6  + \/ 2 *\/  3 - 2*\/ 2  /| + 2*\/ 3 *cos|-- + 2*atan\\/ 3  + \/    1 + \\/ 3  - \/ 6  + \/ 2 *\/  3 - 2*\/ 2  /   - \/ 6  + \/ 2 *\/  3 - 2*\/ 2  /|*sin|-- + 2*atan\\/ 3  + \/    1 + \\/ 3  - \/ 6  + \/ 2 *\/  3 - 2*\/ 2  /   - \/ 6  + \/ 2 *\/  3 - 2*\/ 2  /|   
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Entonces
$$x < - 2 \operatorname{atan}{\left(- \sqrt{6} + \sqrt{2} \sqrt{3 - 2 \sqrt{2}} + \sqrt{\left(- \sqrt{6} + \sqrt{2} \sqrt{3 - 2 \sqrt{2}} + \sqrt{3}\right)^{2} + 1} + \sqrt{3} \right)}$$
no se cumple
significa que una de las soluciones de nuestra ecuación será con:
$$x > - 2 \operatorname{atan}{\left(- \sqrt{6} + \sqrt{2} \sqrt{3 - 2 \sqrt{2}} + \sqrt{\left(- \sqrt{6} + \sqrt{2} \sqrt{3 - 2 \sqrt{2}} + \sqrt{3}\right)^{2} + 1} + \sqrt{3} \right)} \wedge x < - 2 \operatorname{atan}{\left(- \sqrt{6} - \sqrt{2} \sqrt{3 - 2 \sqrt{2}} + \sqrt{1 + \left(- \sqrt{3} + \sqrt{2} \sqrt{3 - 2 \sqrt{2}} + \sqrt{6}\right)^{2}} + \sqrt{3} \right)}$$

         _____           _____  
        /     \         /     \  
-------ο-------ο-------ο-------ο-------
       x3      x2      x4      x1

Recibiremos otras soluciones de la desigualdad pasando al polo siguiente etc.
etc.
Respuesta:
$$x > - 2 \operatorname{atan}{\left(- \sqrt{6} + \sqrt{2} \sqrt{3 - 2 \sqrt{2}} + \sqrt{\left(- \sqrt{6} + \sqrt{2} \sqrt{3 - 2 \sqrt{2}} + \sqrt{3}\right)^{2} + 1} + \sqrt{3} \right)} \wedge x < - 2 \operatorname{atan}{\left(- \sqrt{6} - \sqrt{2} \sqrt{3 - 2 \sqrt{2}} + \sqrt{1 + \left(- \sqrt{3} + \sqrt{2} \sqrt{3 - 2 \sqrt{2}} + \sqrt{6}\right)^{2}} + \sqrt{3} \right)}$$
$$x > - 2 \operatorname{atan}{\left(- \sqrt{6} - \sqrt{\left(- \sqrt{6} + \sqrt{2} \sqrt{3 - 2 \sqrt{2}} + \sqrt{3}\right)^{2} + 1} + \sqrt{2} \sqrt{3 - 2 \sqrt{2}} + \sqrt{3} \right)} \wedge x < 2 \operatorname{atan}{\left(- \sqrt{3} + \sqrt{2} \sqrt{3 - 2 \sqrt{2}} + \sqrt{1 + \left(- \sqrt{3} + \sqrt{2} \sqrt{3 - 2 \sqrt{2}} + \sqrt{6}\right)^{2}} + \sqrt{6} \right)}$$