Is this set of functions linearly dependent or independent?

Question

The given functions are solutions to a differential equation

\begin{equation*} y_1(x)=\cos(2x),y_2(x)=1,\;y_3(x)=\cos(x) \end{equation*}

I need help determining if the set of functions are linearly dependent or independent.

Taking the Wronskian:

\begin{equation*} W(y_1, y_2, y_3) = \begin{vmatrix} \cos(2x) & 1 & \cos(x) \\ -2\sin(2x) & 0 & -\sin(x) \\ -4\cos(2x) & 0 & -\cos(x) \end{vmatrix} = -4\sin^3(x) \end{equation*}

I am not sure if this result tells us linear dependence or independence, as some definitions I found online say that if the wronskian is nonzero for some point x, then the set of functions are linearly independent. And some sources say that the wronskian must be nonzero for every x in order for the set of functions to be linearly independent. I am probably missing something here. Any help is appreciated.

Claim 2

Answer 1

A sufficient but not necessary condition for a set of differentiable functions to be linearly independent on an interval is that their Wronskian does not vanish identically. Their dependence depends not only on the functions themselves but also on the interval. In your example, these functions are independent if the interval is not degenerate.