(x +iy) sq = (a + ib) prove that a sq + b sq =( x sq + y sq )sq
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(x+iy)^2
=x^2-y^2+2ixy
Given (x+iy)^2=a+ib
x^2-y^2+2ixy=a+it
By comparing on both sides
a=x^2-y^2 ,b=2x y
a^2+b^2=(x^2-y^2)^2+(2xy)^2
=x^4+y^4-2(x^2)(y^2)+4 (x^2)
(y^2)
=x^4+y^4+2 (x^2)(y^2)
=(x^2+y^2)^2
Hence proved
=x^2-y^2+2ixy
Given (x+iy)^2=a+ib
x^2-y^2+2ixy=a+it
By comparing on both sides
a=x^2-y^2 ,b=2x y
a^2+b^2=(x^2-y^2)^2+(2xy)^2
=x^4+y^4-2(x^2)(y^2)+4 (x^2)
(y^2)
=x^4+y^4+2 (x^2)(y^2)
=(x^2+y^2)^2
Hence proved
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