Find the value of `sin^(-1)(cos(sin^(-1)x))+cos^(-1)(sin(cos^(-1)x))`

Explanation

To find the value of the expression \( \sin^{-1}(\cos(\sin^{-1}x)) + \cos^{-1}(\sin(\cos^{-1}x)) \), we will break it down step by step.

### Step 1: Define the angles
Let:
- \( \theta = \sin^{-1}(x) \)
- \( \phi = \cos^{-1}(x) \)

From these definitions, we have:
- \( \sin(\theta) = x \)
- \( \cos(\phi) = x \)

### Step 2: Find \( \cos(\theta) \)
Using the Pythagorean identity, we can find \( \cos(\theta) \):
\[ \cos(\theta) = \sqrt{1 - \sin^2(\theta)} = \sqrt{1 - x^2} \]

### Step 3: Find \( \sin(\phi) \)
Again using the Pythagorean identity, we can find \( \sin(\phi) \):
\[ \sin(\phi) = \sqrt{1 - \cos^2(\phi)} = \sqrt{1 - x^2} \]

### Step 4: Substitute back into the original expression
Now we substitute these values back into the original expression:
\[ \sin^{-1}(\cos(\sin^{-1}x)) + \cos^{-1}(\sin(\cos^{-1}x)) = \sin^{-1}(\cos(\theta)) + \cos^{-1}(\sin(\phi)) \]
This simplifies to:
\[ \sin^{-1}(\sqrt{1 - x^2}) + \cos^{-1}(\sqrt{1 - x^2}) \]

### Step 5: Use the identity
We know that:
\[ \sin^{-1}(y) + \cos^{-1}(y) = \frac{\pi}{2} \]
for any \( y \) in the range \([-1, 1]\). Here, \( y = \sqrt{1 - x^2} \) is valid since \( x \) is in the range \([-1, 1]\).

### Final Result
Thus, we conclude that:
\[ \sin^{-1}(\cos(\sin^{-1}x)) + \cos^{-1}(\sin(\cos^{-1}x)) = \frac{\pi}{2} \]