Chapter 6 - Analytic Trigonometry - Section 6.2 The Inverse Trigonometric Functions (Continued) - 6.2 Assess Your Understanding - Page 480: 15

$\sqrt 2$

The range of $\tan^{-1}{x}$ is $[-\dfrac{\pi}{2},\dfrac{\pi}{2}]$. Here, $\tan^{-1}(1)=\dfrac{\pi}{4}$, because $\tan{(\dfrac{\pi}{4})}=1$ and $\dfrac{\pi}{4}$ lies in the range of $\tan^{-1}{x}$ . Therefore, we have: $\csc [\tan^{-1}(1)]=\csc ({\dfrac{\pi}{4}})=\sqrt 2$