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Question
The argument of \[\frac{1 - i}{1 + i}\] is
Options
\[- \frac{\pi}{2}\]
\[\frac{\pi}{2}\]
\[\frac{3\pi}{2}\]
\[\frac{5\pi}{2}\]
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Solution
\[- \frac{\pi}{2}\]
\[\text { Let } z = \frac{1 - i}{1 + i}\]
\[ \Rightarrow z=\frac{1 - i}{1 + i}\times\frac{1 - i}{1 - i}\]
\[ \Rightarrow z=\frac{1 + i^2 - 2i}{1 - i^2}\]
\[ \Rightarrow z = \frac{1 - 1 - 2i}{1 + 1}\]
\[ \Rightarrow z=\frac{- 2i}{2}\]
\[ \Rightarrow z= - i\]
\[\text { Since, z lies on negative direction of imaginary axis } . \]
\[\text { Therefore, } \arg (z) = \frac{- \pi}{2}\]
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