The value of ${\cos }^{-1}\left(\cos \left(-{680}^{\circ }\right)\right)$,is

${\cos }^{-1}\left\{\cos \left(-{680}^{\circ }\right)\right\}={\cos }^{-1}\left(\cos {680}^{\circ }\right)$

$\begin{array}{l}={\cos }^{-1}\left\{\cos \left(680\times \frac{\pi }{180}\right)\right\}={\cos }^{-1}\left(\cos \frac{34\pi }{9}\right)\\ ={\cos }^{-1}\left\{\cos \left(4\pi -\frac{2\pi }{9}\right)\right\}={\cos }^{-1}\left(\cos \frac{2\pi }{9}\right)=\frac{2\pi }{9}\end{array}$