Which of these has the greatest principal value?

Explanation

1. 01

   $\cot^{-1}\left(-\sqrt{3}\right) = \frac{5\pi}{6}$, $\cos^{-1}\left(-\frac{1}{2}\right) = \frac{2\pi}{3}$ and $\sin^{-1}(1) = \frac{\pi}{2}$.

2. 02

   $\tan^{-1}(1000)$ stays below $\frac{\pi}{2}$, however large the input; so $\frac{5\pi}{6}$ is the greatest.