${\rm{cose}}{{\rm{c}}^{ - 1}}(2)$

Let ${\rm{cose}}{{\rm{c}}^{ - 1}}(2) = x \Rightarrow 2 = {\rm{cosec}}\;{\rm{x}}$

As we know that the range of the principal value branch of ${\rm{cose}}{{\rm{c}}^{ - 1}}\;\;is\;\;\left[ { - \frac{\pi }{2},\;\frac{\pi }{2}} \right] - \{ 0\}$

Then, $2 = {\rm{cosec}}\;{\rm{x}}\;{\rm{ = cosec }}\left( {\frac{\pi }{6}} \right),\;where\;\frac{\pi }{6} \in \left[ {\frac{{ - \pi }}{2},\;\frac{\pi }{2}} \right] - \{ 0\}$

Hence, the principal value of ${\rm{cose}}{{\rm{c}}^{ - 1}}(2)\;is\;\pi /6.$