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Question
If \[z = \frac{1}{1 - cos\theta - i sin\theta}\] then Re (z) =
Options
0
\[\frac{1}{2}\]
\[\cot\frac{\theta}{2}\]
\[\frac{1}{2}\cot\frac{\theta}{2}\]
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Solution
\[\frac{1}{2}\]
\[z = \frac{1}{1 - \cos\theta - i\sin\theta}\]
\[z = \frac{1}{1 - \cos\theta - i\sin\theta} \times \frac{1 - \cos\theta + i\sin\theta}{1 - \cos\theta + i\sin\theta}\]
\[ \Rightarrow z=\frac{1 - \cos\theta + i\sin\theta}{\left( 1 - \cos\theta \right)^2 - \left( i\sin\theta \right)^2}\]
\[ \Rightarrow z=\frac{1 - \cos\theta + i\sin\theta}{1 + \cos^2 \theta - 2\cos\theta + \sin^2 \theta}\]
\[ \Rightarrow z= \frac{1 - \cos\theta + i\sin\theta}{1 + 1 - 2\cos\theta}$\]
\[ \Rightarrow z=\frac{1 - \cos\theta + i\sin\theta}{2(1 - \cos\theta)}\]
\[ \Rightarrow \text { Re }(z)=\frac{\left( 1 - \cos\theta \right)}{2\left( 1 - \cos\theta \right)}=\frac{1}{2}\]
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