Explanation

(b)
Let $\mathcal{z =}x + iy$
$\therefore\ \arg{\left( \mathcal{z} \right) = \tan^{- 1}\left( \frac{y}{x} \right)}$
Then, $\arg{\left( \overline{\mathcal{z}} \right) = \tan^{- 1}{\left( - \frac{y}{x} \right) = 2\pi - \tan^{- 1}\frac{y}{x}}}$
$= 2\pi - \arg\left( \mathcal{z} \right)$
Since, in argument of a conjugate of a complex, the real axis is unaltered, but imaginary axis be changed, hence it is given by $2\pi - \arg\left( \mathcal{z} \right)$