Solution

Let \(cot^{-1}(\frac{-1}{\sqrt{3}}) = y\). Then, \(cot~y = \frac{-1}{\sqrt{3}} = -cot(\frac{\pi}{3}) = cot(\pi - \frac{\pi}{3}) = cot(\frac{2\pi}{3})\) We know that the range of principal value branch of \(cot^{-1}\) is \((0, \pi)\) and \(cot(\frac{2\pi}{3}) = \frac{-1}{\sqrt{3}}\). Hence, principal value of \(cot^{-1}(\frac{-1}{\sqrt{3}})\) is \(\frac{2\pi}{3}\).