Question

1/1-costheta-isintheta​

Answer 1

$\bf\ \frac{1}{(1 - cos \theta) - i \: sin \theta} \\ \\ \bf\ \hookrightarrow \frac{1(1 - cos \theta) + i \: sin \theta}{((1 - cos \theta) - i \: sin \theta)((1 - cos \theta) + i \: sin \theta)} \\ \\ \bf\ \hookrightarrow \: \frac{(1 - cos \theta) + i \: sin \theta}{( {1 - cos \theta})^{2} - {(i \: sin \theta)}^{2} } \\ \\ \bf\ \hookrightarrow \: \frac{(1 - cos \theta) + i \: sin \theta}{ {1}^{2} + {cos}^{2} \theta - 2sin \theta - ( { - sin}^{2} \theta) } \: \: \: \: \: ( \therefore {i}^{2} = - 1) \\ \\ \bf\ \hookrightarrow \frac{(1 - cos \theta) + i \: sin \theta}{1 + {sin}^{2} \theta + {cos}^{2} \theta - 2sin \theta } \\ \\ \bf\pink{ \underline{ \bigstar \: \: {sin}^{2} \theta + {cos}^{2} \theta = 1 \: \: \bigstar}} \\ \\ \bf\ \: \hookrightarrow \frac{(1 - cos \theta) + i \: sin \theta}{1 + 1 - 2sin \theta} \\ \\ \hookrightarrow \bf\ \frac{(1 - cos \theta) + i \: sin \theta}{2 - 2sin \theta} \\ \\ \hookrightarrow \bf\ \frac{(1 - cos \theta) + i \: sin \theta}{2(1 - cos \theta)} \\ \\ \large\underline{ \underline{ \red{ \mapsto \: \frac{i \: sin \theta}{2} }}} \\ \\ \bf\ \boxed{ \red{a + ib \: form a = 0\: and \: b = \frac{sin \theta}{2}}}$

$\large\boxed{ \fcolorbox{blue}{pink}{hope \: it \: help \: you}}$