Answer
$\frac{\sqrt{5}}{2}$
Work Step by Step
Assume that $\cos\theta =\frac{2}{3}$, then by solving the triangle, we have:
$$\tan \theta=\frac{\sqrt{5}}{2}.$$
Now, since $\theta =\cos^{-1}\frac{2}{3} $, we have
$$\tan(\cos^{-1}\frac{2}{3})=\tan\theta=\frac{\sqrt{5}}{2}.$$
