Give the exact value of the expression without using a calculator. \[ \sin \left(2 \cos ^{-1} \frac{1}{5}\right) \] \[ \sin \left(2 \cos ^{-1} \frac{1}{5}\right)= \] (Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression.)

Step 1 ：The given expression is in the form of \(\sin(2\theta)\), where \(\theta = \cos^{-1}(\frac{1}{5})\). We know that \(\sin(2\theta) = 2\sin(\theta)\cos(\theta)\). So, we need to find the values of \(\sin(\theta)\) and \(\cos(\theta)\). We know that \(\cos(\theta) = \frac{1}{5}\). We can find \(\sin(\theta)\) using the Pythagorean identity \(\sin^2(\theta) + \cos^2(\theta) = 1\).

Step 2 ：Let's find the value of \(\sin(\theta)\). We know that \(\cos(\theta) = \frac{1}{5}\), so \(\sin^2(\theta) = 1 - \cos^2(\theta) = 1 - \left(\frac{1}{5}\right)^2 = \frac{24}{25}\). Therefore, \(\sin(\theta) = \sqrt{\frac{24}{25}} = \frac{2\sqrt{6}}{5}\).

Step 3 ：Now, we can find the value of \(\sin(2\theta)\) using the formula \(\sin(2\theta) = 2\sin(\theta)\cos(\theta)\). Substituting the values of \(\sin(\theta)\) and \(\cos(\theta)\) we get \(\sin(2\theta) = 2 * \frac{2\sqrt{6}}{5} * \frac{1}{5} = \frac{4\sqrt{6}}{25}\).

Step 4 ：Final Answer: The exact value of the expression \(\sin \left(2 \cos ^{-1} \frac{1}{5}\right)\) is \(\boxed{\frac{4\sqrt{6}}{25}}\).