-2sin2(x)<3 inequation

Given the inequality:
$$- 2 \sin^{2}{\left(x \right)} < 3$$
To solve this inequality, we must first solve the corresponding equation:
$$- 2 \sin^{2}{\left(x \right)} = 3$$
Solve:
Given the equation
$$- 2 \sin^{2}{\left(x \right)} = 3$$
transform
$$\cos{\left(2 x \right)} - 4 = 0$$
$$- 2 \sin^{2}{\left(x \right)} - 3 = 0$$
Do replacement
$$w = \sin{\left(x \right)}$$
This equation is of the form

a*w^2 + b*w + c = 0

A quadratic equation can be solved
using the discriminant.
The roots of the quadratic equation:
$$w_{1} = \frac{\sqrt{D} - b}{2 a}$$
$$w_{2} = \frac{- \sqrt{D} - b}{2 a}$$
where D = b^2 - 4*a*c - it is the discriminant.
Because
$$a = -2$$
$$b = 0$$
$$c = -3$$
, then

D = b^2 - 4 * a * c = 

(0)^2 - 4 * (-2) * (-3) = -24

Because D<0, then the equation
has no real roots,
but complex roots is exists.

w1 = (-b + sqrt(D)) / (2*a)

w2 = (-b - sqrt(D)) / (2*a)

or
$$w_{1} = - \frac{\sqrt{6} i}{2}$$
$$w_{2} = \frac{\sqrt{6} i}{2}$$
do backward replacement
$$\sin{\left(x \right)} = w$$
Given the equation
$$\sin{\left(x \right)} = w$$
- this is the simplest trigonometric equation
This equation is transformed to
$$x = 2 \pi n + \operatorname{asin}{\left(w \right)}$$
$$x = 2 \pi n - \operatorname{asin}{\left(w \right)} + \pi$$
Or
$$x = 2 \pi n + \operatorname{asin}{\left(w \right)}$$
$$x = 2 \pi n - \operatorname{asin}{\left(w \right)} + \pi$$
, where n - is a integer
substitute w:
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(w_{1} \right)}$$
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(- \frac{\sqrt{6} i}{2} \right)}$$
$$x_{1} = 2 \pi n - i \operatorname{asinh}{\left(\frac{\sqrt{6}}{2} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(w_{2} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(\frac{\sqrt{6} i}{2} \right)}$$
$$x_{2} = 2 \pi n + i \operatorname{asinh}{\left(\frac{\sqrt{6}}{2} \right)}$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(w_{1} \right)} + \pi$$
$$x_{3} = 2 \pi n + \pi - \operatorname{asin}{\left(- \frac{\sqrt{6} i}{2} \right)}$$
$$x_{3} = 2 \pi n + \pi + i \operatorname{asinh}{\left(\frac{\sqrt{6}}{2} \right)}$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(w_{2} \right)} + \pi$$
$$x_{4} = 2 \pi n + \pi - \operatorname{asin}{\left(\frac{\sqrt{6} i}{2} \right)}$$
$$x_{4} = 2 \pi n + \pi - i \operatorname{asinh}{\left(\frac{\sqrt{6}}{2} \right)}$$
$$x_{1} = - i \operatorname{asinh}{\left(\frac{\sqrt{6}}{2} \right)}$$
$$x_{2} = i \operatorname{asinh}{\left(\frac{\sqrt{6}}{2} \right)}$$
$$x_{3} = \pi - i \operatorname{asinh}{\left(\frac{\sqrt{6}}{2} \right)}$$
$$x_{4} = \pi + i \operatorname{asinh}{\left(\frac{\sqrt{6}}{2} \right)}$$
Exclude the complex solutions:
This equation has no roots,
this inequality is executed for any x value or has no solutions
check it
subtitute random point x, for example

x0 = 0

$$- 2 \sin^{2}{\left(0 \right)} < 3$$

0 < 3

so the inequality is always executed