[tex]\it x\in\Big(\dfrac{3\pi}{2},\ 2\pi\Big) \Rightarrow \dfrac{x}{2}\in \Big(\dfrac{3\pi}{4},\ \pi\Big) \Rightarrow cos\dfrac{x}{2}<0\ \ \ \ \ (1)\\ \\ \\ cosx=\dfrac{1}{9}\ \ \ \ \ (2)\\ \\ \\ cosx=cos2\cdot\dfrac{x}{2}=2cos^2\dfrac{x}{2}-1 \Rightarrow 2cos^2\dfrac{x}{2}=1+cosx\ \stackrel{(2)}{\Longrightarrow}\ 2cos^2\dfrac{x}{2}=1+\dfrac{1}{9} \Rightarrow[/tex]
[tex]\it \Rightarrow 2cos^2\dfrac{x}{2}=\dfrac{10}{9}|_{:2} \Rightarrow cos^2\dfrac{x}{2}=\dfrac{5}{9}\ \stackrel{(1)}{\Longrightarrow}\ cos\dfrac{x}{2}=\ -\dfrac{\sqrt5}{3}[/tex]
