If `sin^(-1)x+sin^(-1)(1-x)=cos^(-1)x`, then `x` belongs to

Options

A. `{1,0}`
B. `{-1,1}`
C. `{0,1/2}`
D. `{2,0}`

Explanation

To solve the equation \( \sin^{-1} x + \sin^{-1} (1 - x) = \cos^{-1} x \), we will proceed step-by-step.

### Step 1: Rewrite the equation
We know that \( \cos^{-1} x = \frac{\pi}{2} - \sin^{-1} x \). Thus, we can rewrite the equation as:
\[ \sin^{-1} x + \sin^{-1} (1 - x) = \frac{\pi}{2} - \sin^{-1} x \]

### Step 2: Rearrange the equation
Now, we can rearrange the equation to isolate \( \sin^{-1} (1 - x) \):
\[ \sin^{-1} (1 - x) = \frac{\pi}{2} - 2 \sin^{-1} x \]

### Step 3: Apply the sine function
Taking the sine of both sides, we have:
\[ 1 - x = \sin\left(\frac{\pi}{2} - 2 \sin^{-1} x\right) \]
Using the identity \( \sin\left(\frac{\pi}{2} - \theta\right) = \cos(\theta) \), we can rewrite this as:
\[ 1 - x = \cos(2 \sin^{-1} x) \]

### Step 4: Use the double angle formula
Using the double angle formula for cosine, we know:
\[ \cos(2\theta) = 1 - 2\sin^2(\theta) \]
Thus, substituting \( \theta = \sin^{-1} x \):
\[ 1 - x = 1 - 2x^2 \]

### Step 5: Simplify the equation
Now, we can simplify this equation:
\[ 1 - x = 1 - 2x^2 \]
Subtracting 1 from both sides gives:
\[ -x = -2x^2 \]
Multiplying both sides by -1 results in:
\[ x = 2x^2 \]

### Step 6: Rearranging the equation
Rearranging gives:
\[ 2x^2 - x = 0 \]
Factoring out \( x \):
\[ x(2x - 1) = 0 \]

### Step 7: Solving for x
Setting each factor to zero gives us:
1. \( x = 0 \)
2. \( 2x - 1 = 0 \) which simplifies to \( x = \frac{1}{2} \)

### Final Step: Determine the range of x
The values of \( x \) that satisfy the original equation are \( x = 0 \) and \( x = \frac{1}{2} \). Therefore, \( x \) belongs to the set:
\[ \{0, \frac{1}{2}\} \]