Question

Let $z=\frac{\cos \theta +i\sin \theta }{\cos \theta -i\sin \theta }$, $\frac{\pi }{4}<0<\frac{\pi }{2}$. Then arg z is

• A. $2\theta$
• B. $2\theta -\pi$
• C. $\pi +2\theta$
• D. None of these

Answer 1

The correct option is A $2\theta$

Converting to Euler's form, we get
$z=\frac{e^{i\theta }}{e^{-i\theta }}$
$=e^{i\theta -(-i\theta )}$
$=e^{i\left(2\theta \right)}$
$=cos2\theta +isin2\theta$
$\Rightarrow arg\left(z\right)=2\theta$
Hence, option 'A' is correct.