8. (2004 National College Entrance Examination - Hunan Paper) Given: $\tan \left(\frac{\pi}{4}+\alpha\right)=2$, find the value of $\frac{1}{2 \sin \alpha \cos \alpha+\cos ^{2} \alpha}$.
8. $\frac{2}{3} \tan \alpha=\tan \left[\left(\alpha+\frac{\pi}{4}\right)-\frac{\pi}{4}\right]=\frac{1}{3} \quad$ So $\quad \frac{1}{2 \sin \alpha \cos \alpha+\cos ^{2} \alpha}=\frac{\sin ^{2} \alpha+\cos ^{2} \alpha}{2 \sin \alpha \cos \alpha+\cos ^{2} \alpha}=\frac{\tan ^{2} \alpha+1}{2 \tan \alpha+1}$
$$
=\frac{2}{3}
$$