`sin^(-1)(sinx)=x` if

Options

A. `x in R`
B. `x in [-1,1]`
C. `x in [-pi/2,pi/2]`
D. `x in [-pi,pi]`

Answer

To solve the equation \( \sin^{-1}(\sin x) = x \), we need to determine the conditions under which this equality holds true.

### Step-by-Step Solution:

1. **Understanding the Inverse Sine Function**:
The function \( \sin^{-1}(y) \) (also known as arcsin) is defined for \( y \) in the range \([-1, 1]\) and produces outputs in the range \([- \frac{\pi}{2}, \frac{\pi}{2}]\).

2. **Domain of the Sine Function**:
The sine function, \( \sin x \), can take any real number as input, but its output is restricted to the interval \([-1, 1]\). Therefore, \( \sin^{-1}(\sin x) \) will yield \( x \) only when \( x \) falls within the range of the arcsin function.

3. **Condition for Equality**:
For \( \sin^{-1}(\sin x) = x \) to hold true, \( x \) must be within the range of \( \sin^{-1} \), which is \([- \frac{\pi}{2}, \frac{\pi}{2}]\). Thus, we need:
\[ -\frac{\pi}{2} \leq x \leq \frac{\pi}{2} \]

4. **Considering Other Cases**:
If \( x \) is outside this interval, the output of \( \sin^{-1}(\sin x) \) will not equal \( x \). Instead, we can express \( \sin^{-1}(\sin x) \) in terms of \( x \) as follows:
- If \( x \) is in the interval \([- \frac{\pi}{2}, \frac{\pi}{2}]\), then \( \sin^{-1}(\sin x) = x \).
- If \( x \) is outside this interval, we can use the periodic property of the sine function and the definition of the arcsin function to find the equivalent angle within the desired range.

5. **Final Condition**:
Hence, the final condition for \( \sin^{-1}(\sin x) = x \) is:
\[ x \in \left[-\frac{\pi}{2}, \frac{\pi}{2}\right] \]

### Summary:
The equation \( \sin^{-1}(\sin x) = x \) holds true if and only if \( x \) lies within the interval \([- \frac{\pi}{2}, \frac{\pi}{2}]\).