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Question

If x2+y2=1, then the value of 1+x+iy1+xiy is

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Solution

Since x2+y2=1
So let x=cosθ and y=sinθ
1+x+iy1+xiy=(1+cosθ)+isinθ(1+cosθ)isinθ

=2cos2θ2+2isinθ2cosθ22cos2θ22isinθ2cosθ2

=cosθ2+isinθ2cosθ2isinθ2

=cosθ2+isinθ2cosθ2isinθ2× cosθ2+isinθ2cosθ2+isinθ2

=(cosθ2+isinθ2)2cos2θ2i2sin2θ2

=cos2θ2+i2sin2θ2+2icosθ2cosθ2cos2θ2+sin2θ2

=cos2θ2sin2θ2+2icosθ2cosθ21

=cosθ+isinθ=x+iy

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