The value of `cos(sin^(-1)x + cos^(-1)x)`, where `|x| le 1`, is

Explanation

To solve the problem of finding the value of \( \cos(\sin^{-1} x + \cos^{-1} x) \), where \( |x| \leq 1 \), we can follow these steps:

### Step 1: Use the property of inverse trigonometric functions
We know that:
\[ \sin^{-1} x + \cos^{-1} x = \frac{\pi}{2} \]
This property holds true for any \( x \) in the range \( -1 \leq x \leq 1 \).

### Step 2: Substitute the property into the expression
Now, we can substitute this result into our expression:
\[ \cos(\sin^{-1} x + \cos^{-1} x) = \cos\left(\frac{\pi}{2}\right) \]

### Step 3: Evaluate \( \cos\left(\frac{\pi}{2}\right) \)
The value of \( \cos\left(\frac{\pi}{2}\right) \) is:
\[ \cos\left(\frac{\pi}{2}\right) = 0 \]

### Step 4: Conclusion
Thus, the value of \( \cos(\sin^{-1} x + \cos^{-1} x) \) is:
\[ \boxed{0} \]

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