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Question

If $$\left(\dfrac {1+i}{1+i}\right)^{3}-\left(\dfrac {1-i}{1+i}\right)^3=x+iy$$, find $$x$$ and $$y$$


Solution

$$\left (\dfrac {1+i}{1-i}\right)\\=\dfrac {(1+i)}{(1-i)}\times \dfrac {(1+i)}{(1+i)}\\=\dfrac {(1+i)^2}{2}\\=\dfrac {(1+i^2 +2i)}{2}\\=\dfrac {2i}{2}=i$$
Similarity, $$\left (\dfrac {1-i}{1+i}\right)=-i$$
Given expression $$=i^3-(-i)^3 =2i^3 =-2i=0+(-2)i$$.
$$x=0,y=-2$$

Mathematics

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