5. $2 \csc \alpha \quad$ Original expression $=\sqrt{\frac{1-\cos \alpha}{1+\cos \alpha}}+\sqrt{\frac{1+\cos \alpha}{1-\cos \alpha}}=\sqrt{\tan ^{2} \frac{\alpha}{2}}+\sqrt{\cot ^{2} \frac{\alpha}{2}}=\left(\tan \frac{\alpha}{2}+\cot \frac{\alpha}{2}\right)=2 \csc \alpha$
5. Given $f(x)=\sqrt{\frac{1-x}{1+x}}$. If $\alpha \in\left(\frac{\pi}{2}, \pi\right)$, then $f(\cos \alpha)+f(-\cos \alpha)$ can be simplified to