Rewrite $\sin \left(2 \sin ^{-1} \frac{u}{5}\right)$ as an algebraic expression in $u$.
Solution
Step 1 ：We are given the expression \(\sin \left(2 \sin ^{-1} \frac{u}{5}\right)\) and we are asked to rewrite it as an algebraic expression in terms of \(u\).
Step 2 ：We recognize that the given expression is a double angle formula in sine. The double angle formula for sine is \(\sin(2\theta) = 2\sin(\theta)\cos(\theta)\). Here, \(\theta = \sin^{-1}\frac{u}{5}\).
Step 3 ：We can use this formula to rewrite the given expression. However, we need to express \(\cos(\sin^{-1}\frac{u}{5})\) in terms of \(u\).
Step 4 ：We know that \(\sin^2\theta + \cos^2\theta = 1\). So, \(\cos\theta = \sqrt{1 - \sin^2\theta}\). Here, \(\theta = \sin^{-1}\frac{u}{5}\), so \(\sin\theta = \frac{u}{5}\). We can substitute this into the equation to find \(\cos\theta\).
Step 5 ：We find that \(\cos\left(\sin^{-1}\frac{u}{5}\right) = \sqrt{1 - \frac{u^2}{25}}\).
Step 6 ：Now we can substitute \(\sin\theta = \frac{u}{5}\) and \(\cos\theta = \sqrt{1 - \frac{u^2}{25}}\) into the double angle formula \(\sin(2\theta) = 2\sin(\theta)\cos(\theta)\).
Step 7 ：We find that \(\sin \left(2 \sin ^{-1} \frac{u}{5}\right)\) can be rewritten as an algebraic expression in \(u\) as \(\boxed{\frac{2u\sqrt{1 - \frac{u^2}{25}}}{5}}\).
