Chapter 7 - Analytic Trigonometry - 7.2 The Inverse Trigonometric Functions (Continued) - 7.2 Assess Your Understanding - Page 458: 20

$2$

Note that $\cos^{-1}{\left(-\dfrac{\sqrt3}{2}\right)}=\dfrac{5\pi}{6}$, because $\cos{\left(\dfrac{5\pi}{6}\right)}=-\dfrac{\sqrt3}{2}$ and $\dfrac{5\pi}{6}$ is in the range of $\cos^{-1}{x}$, which is $\left[0,\pi\right].$ Thus $\csc{\left(\cos^{-1}{\left(-\dfrac{\sqrt3}{2}\right)}\right)}=\csc{\left(\dfrac{5\pi}{6}\right)}=2$