Write the expression as an algebraic expression in $\mathrm{u}, \mathrm{u}>0$. \[ \sin \left[\cos ^{-1}\left(\frac{2}{\sqrt{u^{2}+4}}\right)\right] \] \[ \sin \left[\cos ^{-1}\left(\frac{2}{\sqrt{u^{2}+4}}\right)\right]= \] (Simplify your answer, including any radicals. Use integers or fractions for any $n$

Step 1 ：Let's start by writing the given expression as a composition of trigonometric functions: \(\sin \left[\cos ^{-1}\left(\frac{2}{\sqrt{u^{2}+4}}\right)\right]\)

Step 2 ：Next, we can use the Pythagorean identity \(\sin^2(x) + \cos^2(x) = 1\) to simplify the expression. If we let \(\cos(x) = \frac{2}{\sqrt{u^2+4}}\), then we can solve for \(\sin(x)\) using the Pythagorean identity.

Step 3 ：Substituting \(\cos(x) = \frac{2}{\sqrt{u^2+4}}\) into the Pythagorean identity, we get \(\sin(x) = \sqrt{1 - \frac{4}{u^2 + 4}}\)

Step 4 ：Simplifying the above expression, we get \(\sin(x) = \sqrt{\frac{u^2}{u^2 + 4}}\)

Step 5 ：Finally, substituting \(\sin(x)\) back into the original expression, we get the simplified algebraic expression in terms of $u$ is \(\sin \left[\cos ^{-1}\left(\frac{2}{\sqrt{u^{2}+4}}\right)\right] = \sqrt{\frac{u^{2}}{u^{2}+4}}\)

Step 6 ：\(\boxed{\sin \left[\cos ^{-1}\left(\frac{2}{\sqrt{u^{2}+4}}\right)\right] = \sqrt{\frac{u^{2}}{u^{2}+4}}}\) is the final answer.