3.085. $1-\frac{1}{1-\sin ^{-1}\left(\frac{\pi}{2}+\alpha\right)}$.

Translate the above text into English, please retain the original text's line breaks and format, and output the translation result directly.

3.085. $1-\frac{1}{1-\sin ^{-1}\left(\frac{\pi}{2}+\alpha\right)}$.
## Solution.

$$
\begin{aligned}
& 1-\frac{1}{1-\sin ^{-1}\left(\frac{\pi}{2}+\alpha\right)}=1-\frac{1}{1-\frac{1}{\sin \left(\frac{\pi}{2}+\alpha\right)}}=1-\frac{1}{1-\frac{1}{\cos \alpha}}=1-\frac{1}{\frac{\cos \alpha-1}{\cos \alpha}}= \\
& =1-\frac{\cos \alpha}{\cos \alpha-1}=\frac{\cos \alpha-1 \cdot \cos \alpha}{\cos \alpha-1}=-\frac{1}{\cos \alpha-1}=-\frac{1}{1-2 \sin ^{2} \frac{\alpha}{2}-1}=\frac{1}{2 \sin ^{2} \frac{\alpha}{2}} .
\end{aligned}
$$

Answer: $0.5 \sin ^{-2} \frac{\alpha}{2}$.