Find the principal value $\mathrm{of} {\tan }^{-1}\sqrt{3}-{\sec }^{-1}(-2).$

We have,  $$\begin{gathered} {\tan }^{-1}\sqrt{3}-{\sec }^{-1}(-2) \\ ={\tan }^{-1}\left(\tan \frac{\mathrm{\pi }}{3}\right)-{\sec }^{-1}\left(\sec \frac{2\mathrm{\pi }}{3}\right) \\ \left[\because \tan \frac{\mathrm{\pi }}{3}=\sqrt{3} \mathrm{and} \sec \frac{2\mathrm{\pi }}{3}=-2\right] \left[\begin{array}{c}\because {\tan }^{-1}\left(\mathrm{tan\theta }\right)=\mathrm{\theta };\forall \mathrm{\theta }\in \left(-\frac{\mathrm{\pi }}{2},\frac{\mathrm{\pi }}{2}\right)\mathrm{and}\\ {\sec }^{-1}\left(\mathrm{sec\theta }\right)=\mathrm{\theta };\forall \mathrm{\theta }\in \left[0,\mathrm{\pi }\right]-\left\{\frac{\mathrm{\pi }}{2}\right\}\end{array}\right] \\ =\frac{\mathrm{\pi }}{3}-\frac{2\mathrm{\pi }}{3}=-\frac{\mathrm{\pi }}{3} \end{gathered}$$