Hướng dẫn giải:

Chọn đáp án C

Ta có: $\sin \frac{x}{2}-\cos \frac{x}{2}=\frac{1}{2}\Rightarrow {{\left( \sin \frac{x}{2}-\cos \frac{x}{2} \right)}^{2}}=\frac{1}{4}\Leftrightarrow {{\sin }^{2}}\frac{x}{2}-2\sin \frac{x}{2}\cos \frac{x}{2}+{{\cos }^{2}}\frac{x}{2}=\frac{1}{4}$

$\Leftrightarrow \sin x=\frac{3}{4}\Rightarrow {{\cos }^{2}}x=1-{{\sin }^{2}}x=\frac{7}{16}$

Do $x\in \left( \frac{\pi }{2};\pi  \right)$ nên $\cos x<0$ $\Rightarrow \cos x=\frac{-\sqrt{7}}{4}\Rightarrow \sin 2x=2\sin x\cos x=\frac{-3\sqrt{7}}{8}$.