Suppose z is a complex number such that z≠−$$1$$, |z|$$=1$$, and arg⁡(z)=$$θLet ω $$=z(1$$−z¯$$)z$$¯$$(1+z),than Re⁡$$($$ω$$)$$ $$)$$ is equal to

Correct option is 3−2sin2⁡(θ/2)

Questions

Suppose $z$ is a complex number such that $z \neq - 1$ , $| z | = 1,$ and $\arg (z) =$ $θ$ Let $ω$ $= \frac{z (1 - \bar{z})}{\bar{z} (1 + z)}$ ,than $Re (ω)$ ) is equal to

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1+cos⁡(θ/2)

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detailed solution

Correct option is C

ω=z−zz¯z¯+z¯z=z−11/z+1=(z−1)zz+1 [∵|z|=1]=z2−1z+1−z−1z+1=z−1+1−z1+zRe⁡(ω)=Re⁡(z)−1+Re⁡1−z1+zBut Re⁡1−z1+z=121−z1+z+1−z1+z=121−z1+z+z−1z+1=0∴ Re⁡(ω)=cos⁡θ−1=−2sin2⁡(θ/2)

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Suppose z is a complex number such that z≠−1, |z|=1, and arg⁡(z)=θLet ω =z(1−z¯)z¯(1+z),than Re⁡(ω) ) is equal to

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Ready to Test Your Skills?

free study material

Suppose $z$ is a complex number such that $z \neq - 1$ , $| z | = 1,$ and $\arg (z) =$ $θ$ Let $ω$ $= \frac{z (1 - \bar{z})}{\bar{z} (1 + z)}$ ,than $Re (ω)$ ) is equal to