<s>[INST] 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 $(-∞, ∞)$?
 [/INST] No calculus needed --
If $a\sin^2 x + b\cos^2 x=0$, then $a$ must be $0$ because that's the only way to make the sum $0$ at $x=\pi/2$ where $\cos^2 x=0$.
Similarly $b=0$ is the only way to make the sum zero at $x=0$.
</s>