Chapter 6 - Analytic Trigonometry - Section 6.2 The Inverse Trigonometric Functions (Continued) - 6.2 Assess Your Understanding - Page 480: 20

$2$

The range of $\cos^{-1}{x}$ is $[0, \pi]$. Here, $\cos^{-1}{(-\dfrac{\sqrt 3}{2})}=\dfrac{ 5\pi}{6}$, because $\cos{(\dfrac{5\pi}{6})}=-\dfrac{\sqrt 3}{2}$ and $\dfrac{5\pi}{6}$ lies in the range of $\cos^{-1}{x}$ . Therefore, we have: $\csc [\cos^{-1}{(\dfrac{-\sqrt 3}{2})}]=\csc ({\dfrac{ 5 \pi}{6}})=2$