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Find the modulus of

Question:

Find the modulus of $\frac{1+i}{1-i}-\frac{1-i}{1+i}$

Solution:

$\frac{1+i}{1-i}-\frac{1-i}{1+i}$

$=\frac{(1+i)(1+i)-(1-i)(1-i)}{(1-i)(1+i)}$

$=\frac{1+i^{2}+2 i-1-i^{2}+2 i}{1^{2}-i^{2}}$

$=\frac{4 i}{2} \quad\left(\because i^{2}=-1\right)$

$=2 i$

$\therefore|2 i|=\sqrt{0^{2}+2^{2}}$

$=2 \quad\left(\because|a+b i|=\sqrt{a^{2}+b^{2}}\right)$

$\Rightarrow\left|\frac{1+i}{1-i}-\frac{1-i}{1+i}\right|=2$

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