11. Let $f(x)$ satisfy $2 f(-\sin x)+3 f(\sin x)=4 \sin x \cos x\left(-\frac{\pi}{4} \leqslant x \leqslant \frac{\pi}{4}\right)$, find the expression for $f(x)$.
11. Since $2 f(-\sin x)+3 f(\sin x)=4 \sin x \cos x$, by substituting $-x$ for $x$ we get $2 f(\sin x)+3 f(-\sin x)=-4 \sin x \cos x$, solving which yields $f(\sin x)=4 \sin x \cos x$.

Given $x \in\left[-\frac{\pi}{4}, \frac{\pi}{4}\right]$, then $\cos x=\sqrt{1-\sin ^{2} x}$, hence $f(\sin x)=4 \sin x \sqrt{1-\sin ^{2} x}$. Therefore, $f(x)=4 x \sqrt{1-x^{2}}, x=\sin \theta \in\left[-\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right]$.