Answer
$\dfrac{4 \pi}{5}$
Work Step by Step
Value of certain composite functions formula:
$f^{-1}(f(x)= \cos^{-1}{(\cos{x})} = x \hspace{25pt} 0 \leq x \leq \pi$
Thus,
$\cos^{-1} {\left[\cos{\left(\dfrac{4 \pi}{5} \right)} \right]} = \dfrac{4 \pi}{5}$