8. 已知 $\tan \alpha=\frac{2}{3}$, 求下列各式的值:

(1) $\frac{\cos \alpha-\sin \alpha}{\cos \alpha+\sin \alpha}+\frac{\cos \alpha+\sin \alpha}{\cos \alpha-\sin \alpha}$;

(2) $\frac{1}{\sin \alpha \cos \alpha}$;

(3) $\sin ^{2} \alpha-2 \sin \alpha \cos \alpha+4 \cos ^{2} \alpha$.
(1) $\frac{\cos \alpha-\sin \alpha}{\cos \alpha+\sin \alpha}+\frac{\cos \alpha+\sin \alpha}{\cos \alpha-\sin \alpha}=\frac{1-\tan \alpha}{1+\tan \alpha}+\frac{1+\tan \alpha}{1-\tan \alpha}=\frac{1-\frac{2}{3}}{1+\frac{2}{3}}+\frac{1+\frac{2}{3}}{1-\frac{2}{3}}=\frac{26}{5}$.

(2) $\frac{1}{\sin \alpha \cos \alpha}=\frac{\sin ^{2} \alpha+\cos ^{2} \alpha}{\sin \alpha \cos \alpha}$

$=\frac{\tan ^{2} \alpha+1}{\tan \alpha}=\frac{13}{6}$.

(3) $\sin ^{2} \alpha-2 \sin \alpha \cos \alpha+4 \cos ^{2} \alpha$
$=\frac{\sin ^{2} \alpha-2 \sin \alpha \cos \alpha+4 \cos ^{2} \alpha}{\sin ^{2} \alpha+\cos ^{2} \alpha}$

$=\frac{\tan ^{2} \alpha-2 \tan \alpha+4}{\tan ^{2} \alpha+1}$

$=\frac{\frac{4}{9}-\frac{4}{3}+4}{\frac{4}{9}+1}=\frac{28}{13}$.