If
\[
x = \sqrt{e^{\sin^{-1} t}} \quad \text{and} \quad
y = \sqrt{e^{\cos^{-1} t}},
\]
then find
\[
\frac{d^2 y}{dx^2}.
\]
Show Hint
When dealing with parametric equations involving inverse trigonometric functions, try to simplify the product or sum of the variables first using fundamental identities like $\sin^{-1}t + \cos^{-1}t = \frac{\pi}{2}$. This can significantly reduce the complexity of differentiation.