6. Solution: Let's find the range of the function $z=g(x)=\cos 2x - 2 \sin x$. The function $g(x)$ is defined for all real numbers. We will make a substitution. Let $t=\sin x$. Then
$z=(1-2t^2)-2t=1-2(t^2+t)=1.5-2(t+0.5)^2$ for $t \in [-1; 1]$, and $E_g=[-3; 1.5]$. The function $u=\frac{\pi}{9} z$ takes all values from the interval $\left[-\frac{\pi}{3}; \frac{\pi}{6}\right]$. The range of the function $f(x)$ coincides with the range of the function $y=\cos u$, where $u \in \left[-\frac{\pi}{3}; \frac{\pi}{6}\right]$. Therefore, $E_f=[0.5; 1]$.
Answer: $E_{f}=[0,5 ; 1]$.