Linear independence of $\sin^2(x)$ and $\cos^2(x)$ The Wronskian for $\sin^2x, \cos^2x$ is
\begin{align}
& \left| \begin{array}{cc} \sin^2 x & \cos^2 x \\ 2\sin x\cos x & -2\cos x\sin x \end{array} \right| \\[8pt]
= {} & -2\sin^2x \cos x \sin x - 2 \cos^2 x \sin x \cos x,
\end{align}
with $x = \frac{π}{6},$ this is $=$ 
$$
-\sqrt{\frac{3}{2}}dx
$$
Does this mean $\sin^2x, \cos^2x$ are linearly independent on the interval from $(-∞, ∞)$?

It suffices to show that the Wronskian is not zero for a single value of $x$. We have: $$W(x) = \begin{vmatrix} \sin^2x & \cos^2x \\ 2\sin x \cos x & -2 \sin x \cos x\end{vmatrix} = -2\sin^3x \cos x - 2\sin x \cos^3 x$$
$$W(x) = -2\sin x \cos x = -\sin(2x)$$
Then, $W(\pi/4) = -1 \neq 0$, so the functions are linearly independent.
