2. If the real number $\alpha$ satisfies $\cos \alpha=\tan \alpha$, then the value of $\frac{1}{\sin \alpha}+\cos ^{4} \alpha$ is $\qquad$
$$
\begin{array}{l}
\cos \alpha=\tan \alpha \Rightarrow \cos ^{2} \alpha=\sin \alpha \Rightarrow \sin ^{2} \alpha+\cos ^{2} \alpha=\sin ^{2} \alpha+\sin \alpha=1 \text {. } \\
\text { Then } \frac{1}{\sin \alpha}+\cos ^{4} \alpha=\frac{1}{\sin \alpha}+\sin ^{2} \alpha=\frac{\sin ^{2} \alpha+\sin \alpha}{\sin \alpha}+\sin ^{2} \alpha \\
=\sin \alpha+1+\sin ^{2} \alpha=2 .
\end{array}
$$