若 $\frac{\cos 2 \alpha}{\sin \left(\alpha-\frac{\pi}{4}\right)}=-\frac{\sqrt{2}}{2}$, 则 $\cos \alpha+\sin \alpha$ 的值为 ( )
[解析] $\frac{\cos 2 \alpha}{\sin \left(\alpha-\frac{\pi}{4}\right)}=\frac{\cos ^{2} \alpha-\sin ^{2} \alpha}{\frac{\sqrt{2}}{2}(\sin \alpha-\cos \alpha)}$

$=\frac{(\cos \alpha+\sin \alpha)(\cos \alpha-\sin \alpha)}{\frac{\sqrt{2}}{2}(\sin \alpha-\cos \alpha)}$

$=-\sqrt{2}(\cos \alpha+\sin \alpha)=-\frac{\sqrt{2}}{2}$.

$\therefore \sin \alpha+\cos \alpha=\frac{1}{2}$.