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Question

If x+iy=11+Cosθ+iSinθ, then show that 4x21=0

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Solution

x+iy=11+cosθ+isinθ
=12cos2θ2+i2sinθ2.cosθ2
=12cosθ2×(1cosθ2+isinθ2)
=0.5sec(θ2).1eiθ2
=0.5sec(θ2)eiθ2
=0.5sec(θ2)(cosθ2isinθ2)
=0.5i0.5tanθ2
Hence
x+iy=0.5i0.5tanθ2
Therefore x=0.5=12
Or
2x=1
Or
4x2=1 or
4x21=0.
Hence proved.

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