If `sin^(-1)(1-x)-2 sin^(-1)x=(pi)/(2)`, then `x` is equal to

Options

A. `0,(1)/(2)`
B. `1,(1)/(2)`
C. `0`
D. `(1)/(2)`

Explanation

To solve the equation \( \sin^{-1}(1-x) - 2 \sin^{-1}(x) = \frac{\pi}{2} \), we can follow these steps:

### Step 1: Substitute Variables
Let \( \theta = \sin^{-1}(x) \). Then, we can rewrite the equation as:
\[ \sin^{-1}(1-x) - 2\theta = \frac{\pi}{2} \]

### Step 2: Isolate \( \sin^{-1}(1-x) \)
Rearranging the equation gives us:
\[ \sin^{-1}(1-x) = \frac{\pi}{2} + 2\theta \]

### Step 3: Apply the Sine Function
Taking the sine of both sides, we have:
\[ 1 - x = \sin\left(\frac{\pi}{2} + 2\theta\right) \]

### Step 4: Use the Sine Addition Formula
Using the sine addition formula, we know that:
\[ \sin\left(\frac{\pi}{2} + 2\theta\right) = \cos(2\theta) \]
Thus, we can rewrite the equation as:
\[ 1 - x = \cos(2\theta) \]

### Step 5: Express \( \cos(2\theta) \)
Using the double angle identity for cosine, we have:
\[ \cos(2\theta) = 1 - 2\sin^2(\theta) \]
Substituting this into our equation gives:
\[ 1 - x = 1 - 2\sin^2(\theta) \]

### Step 6: Simplify the Equation
Cancelling the 1 from both sides results in:
\[ -x = -2\sin^2(\theta) \]
This simplifies to:
\[ x = 2\sin^2(\theta) \]

### Step 7: Substitute Back for \( \sin(\theta) \)
Since \( \sin(\theta) = x \), we substitute back:
\[ x = 2x^2 \]

### Step 8: Rearrange the Equation
Rearranging gives us:
\[ 2x^2 - x = 0 \]

### Step 9: Factor the Equation
Factoring out \( x \):
\[ x(2x - 1) = 0 \]

### Step 10: Solve for \( x \)
Setting each factor to zero gives:
1. \( x = 0 \)
2. \( 2x - 1 = 0 \) → \( x = \frac{1}{2} \)

### Conclusion
Thus, the solutions for \( x \) are:
\[ x = 0 \quad \text{or} \quad x = \frac{1}{2} \]