385. $(\tan \alpha+\cot \alpha)^{2}-(\tan \alpha-\cot \alpha)^{2}$.
Solution. We will use the formulas for the square of the sum and difference of two numbers:

$$
\begin{gathered}
(\operatorname{tg} \alpha+\operatorname{ctg} \alpha)^{2}-(\operatorname{tg} \alpha-\operatorname{ctg} \alpha)^{2}=\operatorname{tg}^{2} \alpha+2 \operatorname{tg} \alpha \operatorname{ctg} \alpha+\operatorname{ctg}^{2} \alpha-\operatorname{tg}^{2} \alpha+ \\
+2 \operatorname{tg} \alpha \operatorname{ctg} \alpha-\operatorname{ctg}^{2} \alpha=4 \operatorname{tg} \alpha \operatorname{ctg} \alpha=4
\end{gathered}
$$

(after combining like terms, we applied identity IV).