Answer
$h(x)$ is in $span(f(x),g(x))$.
Work Step by Step
As $f(x)= \sin^{2}x$ and $g(x) = \cos^{2}x$, using the trigonometric identity $cos(2x) = \cos^{2}x -\sin^{2}x$ we have that
\begin{align*}
h(x) &= \cos(2x)\\
&= \cos^{2} x - \sin^{2}x \\
&= 1\cdot \cos^{2} x + (-1)\cdot \sin^{2}x \\
&=1\cdot g(x)+ (-1)\cdot f(x)
\end{align*}
and clearly $ 1\cdot g(x) + (-1)\cdot f(x) \in span(f(x),g(x))$, so $h(x) \in span(f(x),g(x))$.
