# Find the Modulus of 1 + I 1 − I − 1 − I 1 + I . - Mathematics

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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#### APPEARS IN

RD Sharma Class 11 Mathematics Textbook
Chapter 13 Complex Numbers
Exercise 13.2 | Q 7 | Page 32