If $f(x)=\sin x,-\frac{\pi}{2} \leq x \leq \frac{\pi}{2}$ and $g(x)=\cos x, 0 \leq x \leq \pi$ Find the exact value of the composite function $g\left(f^{-1}\left(\frac{12}{13}\right)\right)$

Step 1 ：First, we need to find the inverse of the function \(f(x)\), which is \(f^{-1}(x)\).

Step 2 ：Then, we substitute \(\frac{12}{13}\) into \(f^{-1}(x)\) to get \(x = 1.1760052070951352\).

Step 3 ：Next, we substitute this result into the function \(g(x)\) to get the final answer.

Step 4 ：The exact value of the composite function \(g\left(f^{-1}\left(\frac{12}{13}\right)\right)\) is approximately 0.38461538461538447.

Step 5 ：Using Python to simplify the final answer, we get \(\boxed{0.385}\).