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Question

If |z1|=1, then the value of tan(arg(z1)2)2iz is

A
1
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B
1
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C
i
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D
i
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Solution

The correct option is D i
Given that |z1|=1
Here z1=cosθ+isinθ
Where θ=arg(z1)
z=1+cosθ+isinθz=2cos2θ2+i2sinθ2cosθ2z=2cosθ2(cosθ2+isinθ2)
From above equation, we get
|z|=2cosθ2, argz=θ2
Hence
tan(arg(z1)2)2iz=tanθ2icosθ2(cosθ2+isinθ2)=tanθ2i(cosθ2isinθ2)cosθ2=i

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