Find the maximum and minimum values of `(sin^(-1)x)^3+(cos^(-1)x)^3,` where `-1lt=xlt=1.`

Explanation

To find the maximum and minimum values of the expression \( A = (\sin^{-1} x)^3 + (\cos^{-1} x)^3 \) for \( -1 < x < 1 \), we can follow these steps:

### Step 1: Use the identity for inverse trigonometric functions
We know that:
\[ \sin^{-1} x + \cos^{-1} x = \frac{\pi}{2} \]
This allows us to express \( \cos^{-1} x \) in terms of \( \sin^{-1} x \):
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