The domain of the function $f\left(x\right)=\cos \left[\log \left\{\frac{\sqrt{9-x^2}}{2-x}\right\}\right]$ is

$$\begin{gathered} \frac{\sqrt{9-x^2}}{2-x}>0\text{\hspace{0.17em}\hspace{0.17em}and\hspace{0.17em}\hspace{0.17em}}9-x^2>0\text{\hspace{0.17em}\hspace{0.17em}and\hspace{0.17em}\hspace{0.17em}}2-x\ne 0 \\ 2-x>0 x^2-9<0\text{\hspace{0.17em}\hspace{0.17em}and\hspace{0.17em}\hspace{0.17em}}x\ne 2 \\ x-2<0 (x+3)(x-3)<0 \\ x<2\to \left(1\right) x\in (-3,3)\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}and\hspace{0.17em}\hspace{0.17em}}x\ne 2\to \left(2\right) \end{gathered}$$

From (1) and (2)

$x\in (-3,2)$