If the function `f(x)=sin^(-1)x+cos^(-1)x` and g(x) are identical, then g(x) can be equal to

Options

A. `sin^(-1)|x|+|cos^(-1)x|`
B. `tan^(-1)x+cot^(-1)x`
C. `|sin^(-1)x|+cos^(-1)|x|`
D. `(sqrt(sin^(-1)x))^(2)+(sqrt(cos^(-1)x))^(2)`

Explanation

To solve the problem, we need to analyze the function \( f(x) = \sin^{-1} x + \cos^{-1} x \) and determine what \( g(x) \) can be equal to if they are identical functions.

### Step-by-Step Solution:

1. **Understanding the Function \( f(x) \)**:
\[ f(x) = \sin^{-1} x + \cos^{-1} x \]
We know from trigonometric identities that:
\[ \sin^{-1} x + \cos^{-1} x = \frac{\pi}{2} \quad \text{for } x \in [-1, 1] \]
Therefore, \( f(x) = \frac{\pi}{2} \) for all \( x \) in the domain \([-1, 1]\).

2. **Identifying the Function \( g(x) \)**:
Since \( g(x) \) is said to be identical to \( f(x) \), it must also equal \( \frac{\pi}{2} \) for \( x \in [-1, 1] \).

3. **Examining the Given Options**:
We need to check which of the options can also equal \( \frac{\pi}{2} \) for \( x \in [-1, 1] \):
- **Option A**: \( \sin^{-1} |x| + \cos^{-1} |x| \)
- **Option B**: \( \tan^{-1} x + \cot^{-1} x \)
- **Option C**: \( \sin^{-1} |x| + \cos^{-1} |x| \)
- **Option D**: \( \sqrt{\sin^{-1} x} + \sqrt{\cos^{-1} x} \)

4. **Evaluating Each Option**:
- **Option A**:
\[ g(x) = \sin^{-1} |x| + \cos^{-1} |x| = \frac{\pi}{2} \quad \text{for } |x| \in [0, 1] \]
This is valid for \( x \in [-1, 1] \).

- **Option B**:
\[ g(x) = \tan^{-1} x + \cot^{-1} x = \frac{\pi}{2} \quad \text{for } x > 0 \]
This does not hold for all \( x \in [-1, 1] \) since it is undefined for \( x = 0 \) and does not equal \( \frac{\pi}{2} \) for negative values.

- **Option C**:
This is the same as Option A, hence it also equals \( \frac{\pi}{2} \) for \( x \in [-1, 1] \).

- **Option D**:
The square roots of inverse sine and cosine do not yield a constant value of \( \frac{\pi}{2} \) for all \( x \in [-1, 1] \).

5. **Conclusion**:
The functions \( g(x) \) that are identical to \( f(x) \) are:
- Option A: \( \sin^{-1} |x| + \cos^{-1} |x| \)
- Option C: \( \sin^{-1} |x| + \cos^{-1} |x| \)

Thus, the correct answer is:
\[ \text{Option C: } \sin^{-1} |x| + \cos^{-1} |x| \]