Prove that `sin^(-1) cos (sin^(-1) x) + cos^(-1) x) = (pi)/(2), |x| le 1`

Explanation

To prove that \( \sin^{-1}(\cos(\sin^{-1}(x))) + \cos^{-1}(x) = \frac{\pi}{2} \) for \( |x| \leq 1 \), we will consider two cases: when \( x \) is in the range \( [0, 1] \) and when \( x \) is in the range \( [-1, 0] \).

### Step 1: Case 1 - When \( x \) is in the range \( [0, 1] \)

1. Start with the left-hand side (LHS):
\[ LHS = \sin^{-1}(\cos(\sin^{-1}(x))) + \cos^{-1}(x) \]
...