$\sin ^{-1} x=\tan ^{-1}\left(\frac{x}{\sqrt{1-x^{3}}}\right)$

Step 1 ：Given the trigonometric equation \(\sin ^{-1} x=\tan ^{-1}\left(\frac{x}{\sqrt{1-x^{3}}}\right)\).

Step 2 ：We need to find the values of x that satisfy the equation.

Step 3 ：By solving this equation, we find that the root is approximately 0, which means x = 0 is a potential solution to the equation.

Step 4 ：However, we need to verify this solution because the domain of the arcsin function is [-1, 1] and the domain of the arctan function is all real numbers.

Step 5 ：The domain of the function under the arctan is also limited because the denominator cannot be 0, which means x cannot be 1.

Step 6 ：Therefore, the solution x = 0 is within the domain of both functions and is a valid solution.

Step 7 ：\(\boxed{\text{Final Answer: The solution to the equation } \sin ^{-1} x=\tan ^{-1}\left(\frac{x}{\sqrt{1-x^{3}}}\right) \text{ is } x = 0}\)