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{\bf Question}

Prove that $$\frac{1+\cos \alpha + i \sin \alpha}{1 - \cos \alpha
+ \sin \alpha} = \cot \frac{1}{2} \alpha \cdot
\exp\left[1\left(\alpha-\frac{\pi}{2}\right)\right]$$

\vspace{.25in}

{\bf Answer}

\begin{eqnarray*} \frac{1+e^{i\alpha}}{1-e^{-i\alpha}}  & = &
\frac{e^{i\frac{\alpha}{2}} \cdot
e^{i\frac{\alpha}{2}}(e^{-i\frac{\alpha}{2}} +
e^{i\frac{\alpha}{2}})}{e^{i\frac{\alpha}{2}} -
e^{-i\frac{\alpha}{2}}} \\ & = & \frac{e^{i{\alpha}} \cot
\frac{\alpha}{2}}{i} \\ & = & \cot \frac{\alpha}{2} e^{i(\alpha -
\frac{\pi}{2})} \end{eqnarray*}

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