3.316. $\sin ^{2}\left(\frac{5}{4} \pi-2 \alpha\right)-\sin ^{2}\left(\frac{5}{4} \pi+2 \alpha\right)$.
## Solution.

$$
\begin{aligned}
& \sin ^{2}\left(\frac{5}{4} \pi-2 \alpha\right)-\sin ^{2}\left(\frac{5}{4} \pi+2 \alpha\right)=\left[\sin ^{2} \frac{x}{2}=\frac{1-\cos x}{2}\right]= \\
& =\frac{1-\cos \left(\frac{5 \pi}{2}-4 \alpha\right)}{2}-\frac{1-\cos \left(\frac{5 \pi}{2}+4 \alpha\right)}{2}= \\
& =\frac{1}{2}\left(\cos \left(\frac{5 \pi}{2}+4 \alpha\right)-\cos \left(\frac{5 \pi}{2}-4 \alpha\right)\right)= \\
& =\frac{1}{2}(-\sin 4 \alpha-\sin 4 \alpha)=-\sin 4 \alpha
\end{aligned}
$$

Answer: $-\sin 4 \alpha$.