Find the principal value of each of the following: ${\tan }^{-1}\left(\frac{1}{\sqrt{3}}\right)$

Find the principal value of each of the following: ${\tan }^{-1}\left(-\frac{1}{\sqrt{3}}\right)$

Find the principal value of each of the following: ${\tan }^{-1}\left(\cos \frac{\pi }{2}\right)$

Find the principal value of each of the following: ${\tan }^{-1}\left(2\cos \frac{2\pi }{3}\right)$

For the principal values, evaluate each of the following: ${\tan }^{-1}(-1)+{\cos }^{-1}\left(-\frac{1}{\sqrt{2}}\right)$

Find the principal values, evaluate each of the following: ${\tan }^{-1}\left\{2\sin \left(4{\cos }^{-1}\frac{\sqrt{3}}{2}\right)\right\}$

Evaluate each of the following: ${\tan }^{-1}1+{\cos }^{-1}\left(-\frac{1}{2}\right)+{\sin }^{-1}\left(-\frac{1}{2}\right)$

Evaluate each of the following: ${\tan }^{-1}\left(-\frac{1}{\sqrt{3}}\right)+{\tan }^{-1}(-\sqrt{3})+{\tan }^{-1}\left(\sin \left(-\frac{\pi }{2}\right)\right)$

Evaluate each of the following: ${\tan }^{-1}\left(\tan \frac{5\pi }{6}\right)+{\cos }^{-1}\left\{\cos \left(\frac{13\pi }{6}\right)\right\}$