Fun with Complex

$\alpha =1+i,\quad \beta =1-i\\ x+\alpha =x+1+i=r{ e }^{ i\Psi }\quad where\quad \tan { \Psi } =\frac { 1 }{ x+1 } \quad \quad ......(1)\\ and\quad r=\sqrt { \left( x+1 \right) ^{ 2 }+1 } =\sqrt { \cot ^{ 2 }{ \Psi } +1 } =\csc { \Psi } \quad ........(2)\\ x+\beta =x+1-i=r{ e }^{ -i\Psi }\\ Put\quad in\quad the\quad given\quad equation,\\ \frac { { r }^{ n }\left( { e }^{ in\Psi }-{ e }^{ -in\Psi } \right) }{ 2i } =\frac { { r }^{ n }\left( 2i\sin { n\Psi } \right) }{ 2i } ={ r }^{ n }\sin { n\Psi } =\frac { \sin { n\Psi } }{ \sin ^{ n }{ \Psi } } \quad \quad .....\left[ from\left( 2 \right) \right] \\ \Longrightarrow \Psi =\theta \quad \quad \quad \quad \Rightarrow \cot { \Psi } =\cot { \theta } \\ \Rightarrow x+1=\cot { \theta } \quad \quad .....\left[ from\left( 1 \right) \right] \\ \Rightarrow x=\cot { \theta } -1$