Write the expression as an algebraic expression in $u, u>0$. \[ \sin \left[\tan ^{-1}\left(\frac{u}{4}\right)\right] \] \[ \sin \left[\tan ^{-1}\left(\frac{u}{4}\right)\right]= \] (Simplify your answer, including any radicals. Use integers or fractions for any numbers in the express

Step 1 ：Given the expression \(\sin \left[\tan ^{-1}\left(\frac{u}{4}\right)\right]\)

Step 2 ：We can simplify it by using the identity \(\sin(\tan^{-1}(x)) = \frac{x}{\sqrt{1+x^2}}\). Here, \(x = \frac{u}{4}\)

Step 3 ：Substitute \(x = \frac{u}{4}\) into the identity, we get \(\frac{u}{4\sqrt{1+\frac{u^2}{16}}}\)

Step 4 ：\(\boxed{\sin \left[\tan ^{-1}\left(\frac{u}{4}\right)\right] = \frac{u}{4\sqrt{1+\frac{u^2}{16}}}}\) is the final simplified expression