Rewrite $\sin (2{\tan }^{-1}u)$ as an algebraic expression in $u$.
$$\sin (2{\tan }^{-1}u)=$$

Step 1 ：We are given the expression $\sin (2{\tan }^{-1}u)$ and we are asked to rewrite it as an algebraic expression in terms of $u$.

Step 2 ：We know that $\sin (2x)=2\sin (x)\cos (x)$. We can use this identity to rewrite the expression.

Step 3 ：We also know that $\tan (x)=\frac{\sin (x)}{\cos (x)}$, so we can express $\sin (x)$ and $\cos (x)$ in terms of $u$ using the right triangle definition of sine and cosine.

Step 4 ：In a right triangle, if $\tan (x)=u$, then we can think of $u$ as the ratio of the opposite side to the adjacent side. We can then use the Pythagorean theorem to find the hypotenuse, which will allow us to find $\sin (x)$ and $\cos (x)$.

Step 5 ：Let's denote the opposite side as $u$, the adjacent side as 1, and the hypotenuse as $\sqrt{u^2+1}$.

Step 6 ：Then, $\sin (x)=\frac{u}{\sqrt{u^2+1}}$ and $\cos (x)=\frac{1}{\sqrt{u^2+1}}$.

Step 7 ：Substituting these values into the identity $\sin (2x)=2\sin (x)\cos (x)$, we get $\sin (2x)=\frac{2u}{u^2+1}$.

Step 8 ：Finally, substituting $x={\tan }^{-1}(u)$ back into the equation, we get $\sin (2{\tan }^{-1}u)=\frac{2u}{u^2+1}$.

Step 9 ：$\boxed{{\displaystyle \sin (2{\tan }^{-1}u)=\frac{2u}{u^2+1}}}$ is the final answer.