$\begin{array}{l} 1)\, \tan\left(2x + \dfrac{\pi}{6}\right) = -\dfrac{\sqrt3}{3}\\ \Leftrightarrow \tan\left(2x + \dfrac{\pi}{6}\right) = \tan\dfrac{5\pi}{6}\\ \Leftrightarrow 2x + \dfrac{\pi}{6} = \dfrac{5\pi}{6} + k\pi\\ \Leftrightarrow 2x = \dfrac{2\pi}{3} + k\pi\\ \Leftrightarrow x = \dfrac{\pi}{3} + k\dfrac{\pi}{2}\,\,\,\,(k \in \Bbb Z)\\\\ 2)\,\tan x = -1\\ \Leftrightarrow x = -\dfrac{\pi}{4} + k\pi \,\,\,\,\,(k \in \Bbb Z)\\\\ 3)\, \cos(3x + 50^o) = \dfrac{1}{2}\\ \Leftrightarrow \cos(3x + 50^o) = \cos60^o\\ \Leftrightarrow \left[\begin{array}{l}3x + 50^o = 60^o + k.360^o\\3x + 50^o = -60^o + k.360^o\end{array}\right.\\ \Leftrightarrow \left[\begin{array}{l}3x = 10^o + k.360^o\\3x = -110^o + k.360^o\end{array}\right.\\ \Leftrightarrow \left[\begin{array}{l}x = \dfrac{10^o}{3} + k.120^o\\x = -\dfrac{110^o}{3} + k.120^o\end{array}\right.\,\,\,\,\,\,(k \in \Bbb Z)\\\\ 4)\,\cot x = -\sqrt3\\ \Leftrightarrow \cot x = \cot\dfrac{5\pi}{6}\\ \Leftrightarrow x = \dfrac{5\pi}{6} + k\pi\,\,\,\,\,(k \in \Bbb Z)\\\\ 5)\,\cot\left(x + \dfrac{\pi}{4}\right) = -\dfrac{\sqrt3}{3}\\ \Leftrightarrow \cot\left(x + \dfrac{\pi}{4}\right) = \cot\dfrac{2\pi}{3}\\ \Leftrightarrow x + \dfrac{\pi}{4} = \dfrac{2\pi}{3} + k\pi\\ \Leftrightarrow x = \dfrac{5\pi}{12} + k\pi\,\,\,\,\,\,(k \in \Bbb Z) \end{array}$