If z is a complex number such that $-\frac{\mathrm{\pi }}{2}\le \arg \mathrm{Z}\le \frac{\mathrm{\pi }}{2}$ then which of the following in equality is true.

$\text{ Let }z=\left|z\right|(\cos \theta +i\sin \theta )\text{ where θ}=\arg \left(z\right),-\frac{\pi }{2}\le \theta \le \frac{\pi }{2}$

$|z-\overline{z}|=\left|z\right||\cos \theta +i\sin \theta -(\cos \theta -i\sin \theta \left)\right|=2\left|z\right||i\sin \theta |$

$⟹|z-\overline{z}|=2\left|z\right|\sin \theta \mid$

$\text{ for }-\frac{\pi }{2}\le \theta \le \frac{\pi }{2}|\sin \theta |\le \left|\theta \right|$

$⟹|z-\overline{z}|\le 2\left|z\right|\left|\theta \right|=\left|z\right||\theta -(-\theta \left)\right|$

$\Rightarrow |\mathrm{z}-\overrightarrow{\mathrm{z}}|\le \left|\mathrm{z}\right||\arg \mathrm{z}-\arg \overrightarrow{\mathrm{z}}|$