Answer
$\dfrac{2}{3}(1- \dfrac{1}{\sqrt 5})$
Work Step by Step
Let us consider $p=\cos^{-1} \dfrac{2}{3}$ and $q=\tan^{-1} \dfrac{1}{2}$
This gives $\cos p=\dfrac{2}{3} \implies \sin p=\sqrt{1-\cos^2 p}=\dfrac{\sqrt 5}{3}$
and $\tan q=\dfrac{1}{2} \implies \sin q=\dfrac{1}{\sqrt 5}$; $\cos q=\dfrac{2}{\sqrt 3}$
Now, $\sin(\cos^{-1} \dfrac{2}{3}-\tan^{-1} \dfrac{1}{2} )=\sin (p- q)$
This implies that
$\sin (p- q)=\sin p \cos q-\cos p \cos q$
$\sin(\cos^{-1} \dfrac{2}{3}-\tan^{-1} \dfrac{1}{2})=(\dfrac{\sqrt 5}{3})(\dfrac{2}{\sqrt 5})-(\dfrac{2}{3})(\dfrac{1}{\sqrt 5})$
Hence, $\sin(\cos^{-1} \dfrac{2}{3}-\tan^{-1} \dfrac{1}{2})=\dfrac{2}{3}(1- \dfrac{1}{\sqrt 5})$
