2.
\(cos\left(x-5\right)=\dfrac{\sqrt{3}}{2}\)
\(\Leftrightarrow cos\left(x-5\right)=cos\left(\dfrac{\pi}{6}\right)\)
\(\Leftrightarrow\left[{}\begin{matrix}x-5=\dfrac{\pi}{6}+k2\pi\\x-5=-\dfrac{\pi}{6}+k2\pi\end{matrix}\right.\)
\(\Rightarrow\left[{}\begin{matrix}x=5+\dfrac{\pi}{6}+k2\pi\\x=5-\dfrac{\pi}{6}+k2\pi\end{matrix}\right.\)
\(\Rightarrow x=\left\{5-\dfrac{11\pi}{6};5-\dfrac{13\pi}{6}\right\}\)
4.
\(\Leftrightarrow cot3x=cot\left(-\dfrac{\pi}{3}\right)\)
\(\Leftrightarrow3x=-\dfrac{\pi}{3}+k\pi\)
\(\Leftrightarrow x=-\dfrac{\pi}{9}+\dfrac{k\pi}{3}\)
\(-\dfrac{\pi}{2}< x< 0\Rightarrow-\dfrac{\pi}{2}< -\dfrac{\pi}{9}+\dfrac{k\pi}{3}< 0\)
\(\Rightarrow-\dfrac{7}{6}< k< \dfrac{1}{3}\)
\(\Rightarrow k=\left\{-1;0\right\}\)
\(\Rightarrow x=\left\{-\dfrac{4\pi}{9};-\dfrac{\pi}{9}\right\}\)
