a. $\cot \left(\dfrac{1}{2}x + \dfrac{\pi}{4}\right) = -1$

$\Leftrightarrow \cot \left(\dfrac{1}{2}x + \dfrac{\pi}{4}\right) = \cot \left(-\dfrac{\pi}{4}\right)$

$\Rightarrow \dfrac{1}{2}x + \dfrac{\pi}{4} = -\dfrac{\pi}{4} + k\pi \quad (k \in \mathbb{Z})$

$\Rightarrow \dfrac{1}{2}x = -\dfrac{\pi}{4} - \dfrac{\pi}{4} + k\pi$

$\Rightarrow \dfrac{1}{2}x = -\dfrac{\pi}{2} + k\pi$

$\Rightarrow x = -\pi + k2\pi \quad (k \in \mathbb{Z})$

b. $\tan (30^\circ - 3x) = \tan 75^\circ$

$\Rightarrow 30^\circ - 3x = 75^\circ + k180^\circ \quad (k \in \mathbb{Z})$

$\Rightarrow -3x = 75^\circ - 30^\circ + k180^\circ$

$\Rightarrow -3x = 45^\circ + k180^\circ$

$\Rightarrow x = -15^\circ - k60^\circ \quad (k \in \mathbb{Z})$