Find the exact value of the expression.
\[
\sin \left(2 \cos ^{-1} \frac{5}{13}\right)
\]

Step 1 ：The given expression is in the form of \(\sin(2\cos^{-1}(x))\). We can use the double angle formula for sine, which is \(\sin(2\theta) = 2\sin(\theta)\cos(\theta)\). Here, \(\theta = \cos^{-1}(\frac{5}{13})\).

Step 2 ：We need to find the values of \(\sin(\theta)\) and \(\cos(\theta)\) to substitute into the formula. We know that \(\cos(\theta) = \frac{5}{13}\).

Step 3 ：We can find \(\sin(\theta)\) using the Pythagorean identity \(\sin^2(\theta) = 1 - \cos^2(\theta)\).

Step 4 ：After finding the values of \(\sin(\theta)\) and \(\cos(\theta)\), we can substitute them into the double angle formula to find the value of the given expression.

Step 5 ：The exact value of the expression \(\sin \left(2 \cos ^{-1} \frac{5}{13}\right)\) is \(\boxed{0.7100591715976332}\).