Write the value of

Let z be a complex number with argument θ.

Then,

$z=r e^{i \theta}$

$\Rightarrow \bar{z}=\overline{r e^{i \theta}}=r e^{-i \theta}$

$\Rightarrow$ argument of $\bar{z}$ is $-\theta$

Thus, $\arg (z)+\arg (\bar{z})=0$.