Write the expression as an algebraic (nontrigonometric) expression in $u, u> 0$.
\[
\tan (\arccos u)
\]
\[
\tan (\arccos u)=
\]
(Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression. Rationalize all denominators.)

Step 1 ：Given the trigonometric expression \(\tan (\arccos u)\), we are asked to express it in terms of \(u\) only.

Step 2 ：We can use the identity \(\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}\).

Step 3 ：Since \(\cos(\arccos u) = u\), we can substitute \(\theta\) with \(\arccos u\) in the above equation.

Step 4 ：We also know that \(\sin(\theta) = \sqrt{1 - \cos^2(\theta)}\). So, substituting \(\theta\) with \(\arccos u\) gives us \(\sin(\arccos u) = \sqrt{1 - u^2}\).

Step 5 ：Substituting these values into the identity for \(\tan(\theta)\) gives us the expression in terms of \(u\) only.

Step 6 ：\(\tan (\arccos u) = \frac{\sqrt{1 - u^2}}{u}\)

Step 7 ：\(\boxed{\tan (\arccos u) = \frac{\sqrt{1 - u^2}}{u}}\)