Answer
$$\frac{{24}}{{25}}$$
Work Step by Step
$$\eqalign{
& \sin \left[ {2{{\cos }^{ - 1}}\left( {\frac{3}{5}} \right)} \right] \cr
& {\text{From the triangle shown below }} \cr
& \cos \theta = \frac{3}{5} \cr
& \theta = {\cos ^{ - 1}}\frac{3}{5} \cr
& {\text{Therefore}} \cr
& \sin \left[ {2{{\cos }^{ - 1}}\left( {\frac{3}{5}} \right)} \right] = \sin 2\theta \cr
& \sin 2\theta = 2\sin \theta \cos \theta \cr
& {\text{From the triangle}} \cr
& 2\sin \theta \cos \theta = 2\left( {\frac{4}{5}} \right)\left( {\frac{3}{5}} \right) \cr
& 2\sin \theta \cos \theta = \frac{{24}}{{25}} \cr} $$
