find maxi of cos(cosx)

Since $-1 \le \cos x \le 1$ and $\cos (-\theta) = \cos \theta$ , $\cos 1 \le \cos (\cos x) \le \cos 0 = 1$ . Therefore the maximum value of $\cos(\cos x)$ is $\boxed 1$ .

Maximum is attained at $x=\dfrac π2$ ; $\cos (\cos \frac π2)=\boxed 1$ . The minimum is attained at $x=0$ , since with increase in $x,\cos x$ decreases and $\cos (\cos x)$ increases and since $\cos x \leq 1<\frac π2,\cos (\cos x)$ can never be negative. The minimum value is $\cos (1)\approx \boxed {0.54}$ .