Statement-I: x$\in$Q1 and cosx + cos3x =cos 2x then $x = {30}^0 or {45}^0$
Statement-II: $x\in (0, 2\pi )$ and Cosecx + 2 = 0 then $x=\frac{7\pi }{6}, \frac{11\pi }{6}$
Statement-III: $x\in [0, 2 \pi ]$ and (2cosx–1)(3+2cosx)=0 then $x=\frac{\pi }{3},\frac{5\pi }{3}$
which of the above statements is correct?

$$\begin{gathered} \mathrm{S}-\mathrm{I} : \\ \cos 3\mathrm{x}+\cos \mathrm{x}=\cos 2\mathrm{x} \\ 2\cos 2\mathrm{x} \cos \mathrm{x}-\cos 2\mathrm{x}=0 \\ \cos 2\mathrm{x}(2\cos \mathrm{x}-1)=0 \\ \cos 2\mathrm{x}=0 \cos \mathrm{x}=\frac{1}{2} \\ 2\mathrm{x}=\frac{\mathrm{\pi }}{2} \mathrm{x}=\frac{\mathrm{\pi }}{3} \\ \mathrm{x}=\frac{\mathrm{\pi }}{4} \end{gathered}$$

$$\begin{gathered} \mathrm{S}-\mathrm{II} : \\ \mathrm{cosec} \mathrm{x}=-2 \\ \mathrm{x}=\frac{7\mathrm{\pi }}{6}, \frac{11\mathrm{\pi }}{6} \\ \mathrm{S}-\mathrm{III} : \\ (2\cos \mathrm{x}-1) (3+2\cos \mathrm{x})=0 \\ \cos =\frac{1}{2}=\cos \frac{\mathrm{\pi }}{3} \\ \mathrm{x}=\frac{\mathrm{\pi }}{3}, \frac{5\mathrm{\pi }}{3} \end{gathered}$$