3.481 Find the minimum value of the expression

$A=\frac{\operatorname{ctg} \alpha-\operatorname{tg} \alpha}{\cos 4 \alpha+1}$ for $0<\alpha<\frac{\pi}{4}$.
## Solution.

$\operatorname{ctg} \alpha-\operatorname{tg} \alpha=\frac{\cos ^{2} \alpha-\sin ^{2} \alpha}{\sin \alpha \cdot \cos \alpha}=\frac{2 \cos 2 \alpha}{\sin 2 \alpha} \Rightarrow A=\frac{2 \cos 2 \alpha}{\sin 2 \alpha \cdot 2 \cos ^{2} 2 \alpha}=\frac{2}{\sin 4 \alpha}$.

Since $0<\alpha<\frac{\pi}{4}$, the minimum value of $A$ is achieved when $\sin 4 \alpha=1$, i.e., when $\alpha=\frac{\pi}{8}$.

Answer: 2 when $\alpha=\frac{\pi}{8}$.