Solution: Let us write \[ f(x) = \sin ^2x \text{ and } g(x) = \cos ^2x. \] To see whether the functions $f(x), g(x)$ are linearly independent, let us consider \[ c_1 f(x) + c_2 g(x) = 0 \quad \forall\ x \in [-2\pi , 2\pi ]. \] Since the equation holds for all $x \in [-2\pi , 2\pi ]$, we take $x = 0$ and $x = \frac{\pi}{2}$. We obtain, \begin{align*} x = 0 & \implies c_1 \sin ^2(0) + c_2 \cos ^2(0) = 0 \implies c_2 = 0 \\ x = \frac{\pi}{2} & \implies c_1 \sin ^2 \left( \frac{\pi}{2} \right) + c_2 \cos ^2 \left( \frac{\pi}{2} \right) = 0 \implies c_1 = 0. \end{align*} Therefore, the set $\mathcal{F} $ is linearly independent in $\mathcal{C} [-2\pi , 2\pi ]$.