Find the exact value of $\frac{2 \tan 15^{\circ}}{\left(1-\tan ^{2} 15^{\circ}\right)}$

Step 1 ：The given expression is in the form of \(\frac{2\tan A}{1-\tan^2 A}\) which is equal to \(\tan 2A\). So, we need to find the value of \(\tan 30^{\circ}\).

Step 2 ：Calculate the value of \(\tan 30^{\circ}\) in radians, which is approximately 0.5235987755982988.

Step 3 ：Find the tangent of this value, which is approximately 0.5773502691896257.

Step 4 ：Final Answer: The exact value of \(\frac{2 \tan 15^{\circ}}{\left(1-\tan ^{2} 15^{\circ}\right)}\) is \(\boxed{0.5773502691896257}\).