386. $\sqrt{\sin ^{2} \alpha(1+\operatorname{ctg} \alpha)+\cos ^{2} \alpha(1+\operatorname{tg} \alpha)}$.

386. $\sqrt{\sin ^{2} \alpha(1+\cot \alpha)+\cos ^{2} \alpha(1+\tan \alpha)}$.
$$
\begin{gathered}
\sqrt{\sin ^{2} \alpha(1+\operatorname{ctg} \alpha)+\cos ^{2} \alpha(1+\operatorname{tg} \alpha)}= \\
=\sqrt{\sin ^{2} \alpha+\sin ^{2} \alpha \operatorname{ctg} \alpha+\cos ^{2} \alpha+\cos ^{2} \alpha \operatorname{tg} \alpha}= \\
=\sqrt{\sin ^{2} \alpha+\cos ^{2} \alpha+\sin ^{2} \alpha \frac{\cos \alpha}{\sin \alpha}+\cos ^{2} \alpha \frac{\sin \alpha}{\cos \alpha}}= \\
=\sqrt{\sin ^{2} \alpha+\cos ^{2} \alpha+2 \sin \alpha \cos \alpha}=\sqrt{(\sin \alpha+\cos \alpha)^{2}}=\sin \alpha+\cos \alpha
\end{gathered}
$$

Solution. We will expand the brackets and then replace $\operatorname{tg} \alpha$ and $\operatorname{ctg} \alpha$ by applying identities II and III:

$$
\begin{gathered}
\sqrt{\sin ^{2} \alpha(1+\operatorname{ctg} \alpha)+\cos ^{2} \alpha(1+\operatorname{tg} \alpha)}= \\
=\sqrt{\sin ^{2} \alpha+\sin ^{2} \alpha \operatorname{ctg} \alpha+\cos ^{2} \alpha+\cos ^{2} \alpha \operatorname{tg} \alpha}= \\
=\sqrt{\sin ^{2} \alpha+\cos ^{2} \alpha+\sin ^{2} \alpha \frac{\cos \alpha}{\sin \alpha}+\cos ^{2} \alpha \frac{\sin \alpha}{\cos \alpha}}= \\
=\sqrt{\sin ^{2} \alpha+\cos ^{2} \alpha+2 \sin \alpha \cos \alpha}=\sqrt{(\sin \alpha+\cos \alpha)^{2}}=\sin \alpha+\cos \alpha
\end{gathered}
$$