Let x=sin-1(12/13), so sin(x)=12/13 and we have to find tan(x).
sin(x)=12/13=opposite/hypotenuse in right triangle.
tan(x)=opposite/adjacent.
adjacent2+opposite2=hypotenuse2 (Pythagoras),
adjacent2+122=132,
adjacent2+144=169,
adjacent2=169-144=25,
adjacent=5,
tan(x)=12/5=2.4, so tan(sin-1(12/13))=2.4.