Animated Solution for Mathematics - Trigonometry: If $\theta \in [-2\pi ,2\pi ]$, then the number of solutions of $2\sqrt{2}{\cos }^2\theta +(2-\sqrt{6})\cos \theta -\sqrt{3}=0$, is equal to:

The given trigonometric equation is:

To simplify this, we substitute $x=\cos \theta$. This transforms the expression into a standard quadratic equation:

We expand the middle term $(2-\sqrt{6})x$ into $2x-\sqrt{6}x$ to facilitate grouping. The equation becomes:

By grouping the terms, we extract common factors:

This leads to the factored form:

Substituting back $\cos \theta$ for $x$, we obtain two fundamental trigonometric equations:

We must determine the number of solutions for $\theta \in [-2\pi ,2\pi ]$. This interval represents two full cycles of the cosine wave.

For the first condition, $\cos \theta =\frac{\sqrt{3}}{2}$, the horizontal line intersects the cosine graph twice in each cycle. Over the interval $[-2\pi ,2\pi ]$, this yields:

For the second condition, $\cos \theta =-\frac{1}{\sqrt{2}}$, the horizontal line also intersects the cosine graph twice in each cycle. Over the same interval, this yields:

By summing the solutions from both conditions, we find the total number of values for $\theta$ that satisfy the original equation.

Total solutions = $4+4=8$.

The total number of solutions in the interval $[-2\pi ,2\pi ]$ is 8.