Chapter 7 - Analytic Trigonometry - 7.2 The Inverse Trigonometric Functions (Continued) - 7.2 Assess Your Understanding - Page 458: 24

$\pi$

Note that $\tan{\left(-\dfrac{\pi}{4}\right)}=-1$. Thus $\cos^{-1}\left(\tan{\left(-\dfrac{\pi}{4}\right)}\right)=\cos^{-1}{\left(-1\right)}=\pi$, because $\cos{\left(\pi\right)}=-1$ and $\pi$ is in the range of $\cos^{-1}{x}$, which is $\left[0,\pi\right].$