The value of $\tan[\frac{1}{2}\cos^{-1}(\frac{\sqrt{5}}{3})]$ is:

$\frac{3-\sqrt{5}}{2}$

Put $\cos^{-1}(\frac{\sqrt{5}}{3})=θ$ to get $\cos θ=\frac{\sqrt{5}}{3}$ and $\sin θ=\frac{2}{3};\tan[\frac{1}{2}\cos^{-1}(\frac{\sqrt{5}}{3})]=\tan(\frac{θ}{2})=\frac{\sin θ}{1+\cos θ}=\frac{2/3}{1+\frac{\sqrt{5}}{3}}=\frac{3-\sqrt{5}}{2}$