Question

If $|z|=1$ then $\frac{1+z}{1+\overline{z}}$ is equal to

• A. $z$
• B. $\overline{z}$
• C. $z+\overline{z}$
• D. None of these

Answer 1

The correct option is A $z$

$\because \left|z\right|=1letz=\cos \theta +i\sin \theta$
$w=\frac{1+z}{1+\overline{z}}=\frac{1+\cos \theta +i\sin \theta }{1+\cos \theta -i\sin \theta }=\frac{2{\cos }^2\frac{\theta }{2}+2i\cos \frac{\theta }{2}\sin \frac{\theta }{2}}{2{\cos }^2\frac{\theta }{2}-2i\cos \frac{\theta }{2}\sin \frac{\theta }{2}}$
$\Rightarrow w=\frac{\cos \frac{\theta }{2}+i\sin \frac{\theta }{2}}{\cos \frac{\theta }{2}-i\sin \frac{\theta }{2}}=\frac{cis\frac{\theta }{2}}{cis\frac{-\theta }{2}}=cis[\frac{\theta }{2}-(-\frac{\theta }{2})]=cis\theta$
$\Rightarrow w=\cos \theta +i\sin \theta$
$\therefore \frac{1+z}{1+\overline{z}}=z$
Hence, option 'A' is correct.