Answer
$\frac{\sqrt{5}}{2}$
Work Step by Step
Let $\cos ^ {–1} (\frac{2}{3}) = \theta$
then, $\cos \theta = \frac{2}{3}$ => $\sec \theta = \frac{1}{\cos \theta} = \frac{3}{2}$
To evaluate, $\tan (\cos ^ {–1} (\frac{2}{3})) = \tan \theta$
We know that,
$\tan \theta= \sqrt{\sec ^2 \theta - 1}$
also $\sec \theta = \frac{1}{\cos \theta} $
Using above relations we get
$\tan \theta = \sqrt{(\frac{3}{2})^2 - 1}$
$\tan \theta = \frac{\sqrt{5}}{2}$
