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

<p>$\because$ ${{1 - i\sin \alpha } \over {1 + 2i\sin \alpha }}$ is purely imaginary</p> <p>$\therefore$ $${{1 - i\sin \alpha } \over {1 + 2i\sin \alpha }} + {{1 + i\sin \alpha } \over {1 - 2i\sin \alpha }} = 0$$</p> <p>$\Rightarrow 1 - 2{\sin ^2}\alpha = 0$</p> <p>$\therefore$ $\alpha = {{5\pi } \over 4},\,{{7\pi } \over 4}$</p> <p>and ${{1 + i\cos \beta } \over {1 - 2i\cos \beta }}$ is purely real</p> <p>$${{1 + i\cos \beta } \over {1 - 2i\cos \beta }} - {{1 - i\cos \beta } \over {1 + 2i\cos \beta }} = 0$$</p> <p>$\Rightarrow \cos \beta = 0$</p> <p>$\therefore$ $\beta = {{3\pi } \over 2}$</p> <p>$\therefore$ $$S = \left\{ {\left( {{{5\pi } \over 2},{{3\pi } \over 2}} \right),\left( {{{7\pi } \over 4},{{3\pi } \over 2}} \right)} \right\}$$</p> <p>${Z_{\alpha \beta }} = 1 - i$ and ${Z_{\alpha \beta }} = - 1 - i$</p> <p>$\therefore$ $$\sum\limits_{(\alpha ,\beta ) \in S} {\left( {i{Z_{\alpha \beta }} + {1 \over {i{{\overline Z }_{\alpha \beta }}}}} \right) = i( - 2i) + {1 \over i}\left[ {{1 \over {1 + i}} + {1 \over { - 1 + i}}} \right]} $$</p> <p>$= 2 + {1 \over i}{{2i} \over { - 2}} = 1$</p>