7. 已知向量 $\boldsymbol{a}=(\cos \alpha, \sin \alpha), \boldsymbol{b}=(\cos \beta, \sin \beta),|\boldsymbol{a}-\boldsymbol{b}|=\frac{2 \sqrt{5}}{5}$.

(1)求 $\cos (\alpha-\beta)$ 的值;

(2)若 $-\frac{\pi}{2}<\beta<0<\alpha<\frac{\pi}{2}$, 且 $\sin \beta=-\frac{5}{13}$, 求 $\sin \alpha$ 的值.
(1) $\because|\boldsymbol{a}-\boldsymbol{b}|=\frac{2 \sqrt{5}}{5}$,
$\therefore \boldsymbol{a}^{2}-2 \boldsymbol{a} \cdot \boldsymbol{b}+\boldsymbol{b}^{2}=\frac{4}{5}$,

又 $\boldsymbol{a}=(\cos \alpha, \sin \alpha), \boldsymbol{b}=(\cos \beta, \sin \beta)$,

$\therefore \boldsymbol{a}^{2}=\boldsymbol{b}^{2}=1, \boldsymbol{a} \cdot \boldsymbol{b}=\cos \alpha \cos \beta+\sin \alpha \sin \beta=\cos (\alpha-\beta)$,

$\therefore \cos (\alpha-\beta)=\frac{3}{5}$.

(2) $\because-\frac{\pi}{2}<\beta<0<\alpha<\frac{\pi}{2}, \quad \therefore 0<\alpha-\beta<\pi$,

由(1)得 $\cos (\alpha-\beta)=\frac{3}{5}, \therefore \sin (\alpha-\beta)=\frac{4}{5}$,

又 $\sin \beta=-\frac{5}{13}, \therefore \cos \beta=\frac{12}{13}$,

$\therefore \sin \alpha=\sin [(\alpha-\beta)+\beta]$

$=\sin (\alpha-\beta) \cos \beta+\cos (\alpha-\beta) \sin \beta$

$=\frac{4}{5} \times \frac{12}{13}+\frac{3}{5} \times\left(-\frac{5}{13}\right)=\frac{33}{65}$.