Question

If $z=r(\cos \theta +i\sin \theta ),$ then the value of $\frac{z}{z}=\frac{\overline{z}}{z}$

• A. $\cos 2\theta$
• B. $2\cos 2\theta$
• C. $2\cos \theta$
• D. $2\sin \theta$
• E. $2\sin 2\theta$

Answer 1

Answer: (B)

We have $z=r(\cos \theta +i\sin \theta )$
$\therefore$ $\overline{z}=r(\cos \theta -i\sin \theta )$
$\therefore$ $\frac{z}{z}+\frac{\overline{z}}{z}$
$=\frac{r(\cos \theta +i\sin \theta )}{r(\cos \theta -i\sin \theta )}+\frac{r(\cos \theta -i\sin \theta )}{r(\cos \theta +i\sin \theta )}$
$=\frac{{(\cos \theta +i\sin \theta )}^2+(\cos \theta -i\sin {\theta }^2)}{{\cos }^2\theta +{\sin }^2\theta }$
$={\cos }^2\theta -{\sin }^2\theta +2i\cos \theta \sin \theta +{\cos }^2\theta$ $-{\sin }^2\theta +2i\cos \theta \sin \theta$
$=2({\cos }^2\theta -{\sin }^2\theta )$
$=2\cos 2\theta$