Mathematics - Complex Number Question with Solution | TestHub

Let$z=\cos \theta +i \sin \theta$

Then,$\frac{1+z}{1+\overline{z}}=\frac{1+\cos \theta +i \sin \theta }{1+\cos \theta -i \sin \theta }$

$=\frac{2 {\cos }^2\frac{\theta }{2}+i·2 \sin \frac{\theta }{2}\cos \frac{\theta }{2}}{2 {\cos }^2\frac{\theta }{2}-i·2 \sin \frac{\theta }{2}\cos \frac{\theta }{2}}$

$=\frac{\cos \frac{\theta }{2}+i \sin \frac{\theta }{2}}{\cos \frac{\theta }{2}-i \sin \frac{\theta }{2}}$

$={\left(\cos \frac{\theta }{2}+\mathrm{i} \sin \frac{\theta }{2}\right)}^2$

$=\cos \theta +\mathrm{i} \sin \theta$

$\therefore arg\left(\frac{1+z}{1+\overline{z}}\right)=arg \left(z\right)$

$=\theta$