$3.456 A=\sin ^{2}\left(\frac{1}{2} \arcsin \frac{4}{5}-2 \operatorname{arctg}(-2)\right)$

$3.456 A=\sin ^{2}\left(\frac{1}{2} \arcsin \frac{4}{5}-2 \arctan(-2)\right)$
Solution. Let $\alpha=\arcsin \frac{4}{5}, \beta=\operatorname{arctg} 2 \Rightarrow \sin \alpha=\frac{4}{5}, \cos \alpha=\frac{3}{5}$, $\sin \frac{\alpha}{2}=\sqrt{\frac{1-\cos \alpha}{2}}=\frac{1}{\sqrt{5}}, \quad \cos \frac{\alpha}{2}=\frac{2}{\sqrt{5}}, \operatorname{tg} \beta=2, \sin 2 \beta=\frac{2 \operatorname{tg} \beta}{1.7 \operatorname{tg}^{2} \beta}=\frac{4}{5}$,
$\cos 2 \beta=\frac{1-\operatorname{tg}^{2} \beta}{1+\operatorname{tg}^{2} \beta}=-\frac{3}{5} \Rightarrow \sin \left(\frac{\alpha}{2}+2 \beta\right)=\sin \frac{\alpha}{2} \cdot \cos 2 \beta+\cos \frac{\alpha}{2} \cdot \sin 2 \beta=\frac{1}{\sqrt{5}} \Rightarrow$

$\Rightarrow A=\sin ^{2}\left(\frac{\alpha}{2}+2 \beta\right)=\frac{1}{5}$.

Answer: $A=\frac{1}{5}$.