Give the exact value of the expression without using a calculator.
\[
\sin \left(2 \tan ^{-1} \frac{15}{8}\right)
\]
\[
\sin \left(2 \tan ^{-1} \frac{15}{8}\right)=
\]
(Use radicals or fractions for any numbers in the expression.)

Step 1 ：Let \(a = 2 \tan^{-1} \frac{15}{8}\). Then \(\tan a = \tan(2 \tan^{-1} \frac{15}{8})\).

Step 2 ：Using the double angle formula for tangent, we have \(\tan a = \frac{2 \cdot \frac{15}{8}}{1 - (\frac{15}{8})^2} = \frac{30}{64 - 225} = -\frac{30}{161}\).

Step 3 ：Construct a right triangle with opposite side -30 and adjacent side 161. The hypotenuse is \(\sqrt{(-30)^2 + 161^2} = \sqrt{26081} = 161.5\).

Step 4 ：Then, \(\sin a = \frac{-30}{161.5} = -\frac{6}{32.3}\) and \(\cos a = \frac{161}{161.5} = \frac{322}{322.3}\).

Step 5 ：Using the double angle formula for sine, we have \(\sin 2a = 2 \sin a \cos a = 2 \cdot -\frac{6}{32.3} \cdot \frac{322}{322.3} = -\frac{3864}{10432.89}\).

Step 6 ：Simplify the fraction to get the final answer: \(\boxed{-\frac{1932}{5216.445}}\).