Answer
The solution in the interval $\left[ 0,2\pi \right)$ is $\pi $.
Work Step by Step
We have to solve the equation on the interval $\left[ 0,2\pi \right)$:
$\begin{align}
& \text{si}{{\text{n}}^{2}}x-2\text{cos}x-2=0 \\
& 1-\text{co}{{\text{s}}^{2}}x-2\cos x-2=0 \\
& -\text{co}{{\text{s}}^{2}}x-2\text{cos}x-1=0 \\
& \text{co}{{\text{s}}^{2}}x+2\text{cos}x+1=0
\end{align}$
Next, we factor:
$\begin{align}
& \left( \text{cos}x+1 \right)\left( \text{cos}x+1 \right)=0 \\
& \text{cos}x+1=0 \\
& \text{cos}x=-1 \\
& x=\pi
\end{align}$
