3.082. $1-\frac{1}{1-\sin ^{-1}\left(2 \alpha+\frac{3}{2} \pi\right)}$.

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3.082. $1-\frac{1}{1-\sin ^{-1}\left(2 \alpha+\frac{3}{2} \pi\right)}$.
Solution.

$$
\begin{aligned}
& 1-\frac{1}{1-\sin ^{-1}\left(2 \alpha+\frac{3}{2} \pi\right)}=1-\frac{1}{1-\frac{1}{\sin \left(2 \alpha+\frac{3 \pi}{2}\right)}}=1-\frac{1}{1-\frac{1}{\sin \left(\frac{3 \pi}{2}+2 \alpha\right)}}= \\
& =1-\frac{1}{1-\frac{1}{-\cos 2 \alpha}}=1-\frac{1}{1+\frac{1}{\cos 2 \alpha}}=1-\frac{1}{\frac{\cos 2 \alpha+1}{\cos 2 \alpha}}=1-\frac{\cos 2 \alpha}{\cos 2 \alpha+1}= \\
& =\frac{\cos 2 \alpha+1-\cos 2 \alpha}{\cos 2 \alpha+1}=\frac{1}{\cos 2 \alpha+1}=\frac{1}{2 \cos ^{2} \alpha-1+1}=\frac{1}{2 \cos ^{2} \alpha} \\
& \text { Hence }: \frac{1}{2 \cos ^{2} \alpha}
\end{aligned}
$$