$3.443 A=\operatorname{ctg} \frac{5 \pi}{4}-2 \sin ^{2}\left(\frac{5 \pi}{2}+\frac{1}{2} \arcsin \frac{2 \sqrt{2}-1}{3}\right)$.

$3.443 A=\cot \frac{5 \pi}{4}-2 \sin ^{2}\left(\frac{5 \pi}{2}+\frac{1}{2} \arcsin \frac{2 \sqrt{2}-1}{3}\right)$.
## Solution.

Let $\alpha=\arcsin \frac{2 \sqrt{2}-1}{3}$, then $0<\alpha<\frac{\pi}{2}, \sin \alpha=\frac{2 \sqrt{2}-1}{3}$, $\cos \alpha=\frac{2 \sqrt[4]{2}}{3} \Rightarrow A=1-2 \sin ^{2}\left(\frac{5 \pi}{2}+\frac{\alpha}{2}\right)=\cos (5 \pi+\alpha)=-\cos \alpha=-\frac{2 \sqrt[4]{2}}{3}$.

Answer: $A=-\frac{2 \sqrt[4]{2}}{3}$.