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Q. If $\frac{(1+i)^{3}}{(1-i)^{3}}-\frac{(1-i)^{3}}{(1+i)^{3}}=x+ iy$

Complex Numbers and Quadratic Equations

Solution:

$\frac{(1+i)^{3}}{(1-i)^{3}}-\frac{(1-i)^{3}}{(1+i)^{3}}=x+i y$
$\Rightarrow \frac{\left(1+i^{2}+2 i\right)^{3}-\left(1+i^{2}-2 i\right)^{3}}{\left(1-i^{2}\right)^{3}}=x+i y$
$\Rightarrow \frac{8 i^{3}+8 i^{3}}{2^{3}}=x+i y$
$ \Rightarrow 2 i^{3}=x+i y$
$ \Rightarrow -2 i=x+i y$
$\Rightarrow x=0, y=-2$