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Question
Find the modulus of \[\frac{1 + i}{1 - i} - \frac{1 - i}{1 + i}\].
Solution
\[\frac{1 + i}{1 - i} - \frac{1 - i}{1 + i}\]
\[ = \frac{\left( 1 + i \right)\left( 1 + i \right) - \left( 1 - i \right)\left( 1 - i \right)}{\left( 1 - i \right)\left( 1 + i \right)}\]
\[ = \frac{1 + i^2 + 2i - 1 - i^2 + 2i}{1^2 - i^2}\]
\[ = \frac{4i}{2} \left ( \because i^2 = - 1 \right)\]
\[ = 2i\]
\[ \therefore \left| 2i \right| = \sqrt{0^2 + 2^2}\]
\[ = 2 \left( \because \left| a + bi \right| = \sqrt{a^2 + b^2} \right)\]
\[ \Rightarrow \left| \frac{1 + i}{1 - i} - \frac{1 - i}{1 + i} \right| = 2\]
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