Let S be the set of all (α, β), π<α, β<2 π, for which the complex number (1-i sin α)/(1+2 i sin α) is purely imaginary and (1+i cos β)/(1-2 i cos β)…

Let S be the set of all $\displaystyle (\alpha, \beta), \pi<\alpha, \beta<2 \pi$, for which the complex number $\displaystyle \frac{1-i \sin \alpha}{1+2 i \sin \alpha}$ is purely imaginary and $\displaystyle \frac{1+i \cos \beta}{1-2 i \cos \beta}$ is purely real. Let $\displaystyle Z_{\alpha \beta}=\sin 2 \alpha+i \cos 2 \beta,(\alpha, \beta) \in S$. Then $\displaystyle \sum_{(\alpha, \beta) \in \mathrm{S}}\left(i Z_{\alpha \beta}+\frac{1}{i \bar{Z}_{\alpha \beta}}\right)$ is equal to: