The value of $\cos [\frac{1}{2}{\cos }^{-1}(\cos (-\frac{14\mathrm{\pi }}{5}))]$ is equal to 

$\begin{array}{rl} & \cos (-\frac{14\mathrm{\pi }}{5})=\cos \frac{14\mathrm{\pi }}{5}=\cos \frac{4\mathrm{\pi }}{5}\\ \mathrm{Here}, & \cos \frac{1}{2}{\cos }^{-1}(\cos \frac{4\mathrm{\pi }}{5})=\cos \frac{2\mathrm{\pi }}{5}\end{array}$