My idea was: if $3 + x^2$ was in the subspace spanned by the other two, then it would be some linear combination of those two. So what I did is I formed the Wronskian, found that it was not identically zero everywhere, and concluded that this meant that $3 + x^2$ was not in the span of the other two. I'm actually doubtful that this will work though.
Can someone please explain whether or not this would work, and if it's wrong, please push me in the right direction as to how to approach this question?
Your approach seems to work, but there are simpler ways to do this : for example, notice that any linear combination of $\cos^2$ and $\sin^2$ is periodic (or bounded), while $3+x^2$ isn't...