1.

\(sin2x=-\dfrac{1}{2}\)

\(\Leftrightarrow\left[{}\begin{matrix}2x=-\dfrac{\pi}{6}+k2\pi\\2x=\dfrac{7\pi}{6}+k2\pi\end{matrix}\right.\)

\(\Leftrightarrow\left[{}\begin{matrix}x=-\dfrac{\pi}{12}+k\pi\\x=\dfrac{7\pi}{12}+k\pi\end{matrix}\right.\) 

\(\Rightarrow x=\left\{\dfrac{7\pi}{12};\dfrac{11\pi}{12}\right\}\)

3.

\(tan\left(2x-15^0\right)=1\)

\(\Rightarrow2x-15^0=45^0+k180^0\)

\(\Rightarrow x=30^0+k90^0\)

\(\Rightarrow x=\left\{-60^0;30^0\right\}\)

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.

\(\Leftrightarrow cot3x=cot\left(-\dfrac{\pi}{3}\right)\)

\(\Leftrightarrow3x=-\dfrac{\pi}{3}+k\pi\)

\(\Leftrightarrow x=-\dfrac{\pi}{9}+\dfrac{k\pi}{3}\)

\(-\dfrac{\pi}{2}< x< 0\Rightarrow-\dfrac{\pi}{2}< -\dfrac{\pi}{9}+\dfrac{k\pi}{3}< 0\)

\(\Rightarrow-\dfrac{7}{6}< k< \dfrac{1}{3}\)

\(\Rightarrow k=\left\{-1;0\right\}\)

\(\Rightarrow x=\left\{-\dfrac{4\pi}{9};-\dfrac{\pi}{9}\right\}\)

\(\Leftrightarrow\left(sinx-3\right)\left(2sinx+1\right)=0\)

\(\Leftrightarrow\left[{}\begin{matrix}sinx=3>1\left(loại\right)\\sinx=-\dfrac{1}{2}\end{matrix}\right.\)

\(\Leftrightarrow\left[{}\begin{matrix}x=-\dfrac{\pi}{6}+k2\pi\\x=\dfrac{7\pi}{6}+k2\pi\end{matrix}\right.\)

\(\Leftrightarrow\left(2cosx-1\right)\left(2cosx-\sqrt{3}\right)=0\)

\(\Leftrightarrow\left[{}\begin{matrix}cosx=\dfrac{1}{2}\\cosx=\dfrac{\sqrt{3}}{2}\end{matrix}\right.\)

\(\Rightarrow x=...\)