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

Options

A. `0,1/2`
B. `0,1/2`
C. 0
D. none of these

Answer

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

### Step 1: Substitute \( \sin^{-1}(x) \)
Let \( \sin^{-1}(x) = \theta \). Then, we have:
\[ x = \sin(\theta) \]
Substituting this into the equation gives:
\[ \sin^{-1}(1 - \sin(\theta)) - 2\theta = \frac{\pi}{2} \]

### Step 2: Rearranging the Equation
Rearranging the equation, we get:
\[ \sin^{-1}(1 - \sin(\theta)) = \frac{\pi}{2} + 2\theta \]

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

### Step 4: Use the Cosine Double Angle Formula
Using the double angle formula for cosine, we know:
\[ \cos(2\theta) = 1 - 2\sin^2(\theta) \]
Substituting this into our equation gives:
\[ 1 - \sin(\theta) = 1 - 2\sin^2(\theta) \]

### Step 5: Simplify the Equation
Cancelling \(1\) from both sides:
\[ -\sin(\theta) = -2\sin^2(\theta) \]
Multiplying through by \(-1\):
\[ \sin(\theta) = 2\sin^2(\theta) \]

### Step 6: Rearranging the Equation
Rearranging gives:
\[ 2\sin^2(\theta) - \sin(\theta) = 0 \]

### Step 7: Factor the Equation
Factoring out \(\sin(\theta)\):
\[ \sin(\theta)(2\sin(\theta) - 1) = 0 \]

### Step 8: Solve for \(\sin(\theta)\)
Setting each factor to zero gives:
1. \( \sin(\theta) = 0 \)
2. \( 2\sin(\theta) - 1 = 0 \) which simplifies to \( \sin(\theta) = \frac{1}{2} \)

### Step 9: Find Corresponding \(x\) Values
From \( \sin(\theta) = 0 \):
\[ x = 0 \]

From \( \sin(\theta) = \frac{1}{2} \):
\[ x = \frac{1}{2} \]

### Step 10: Validate the Solutions
1. For \( x = 0 \):
\[ \sin^{-1}(1-0) - 2\sin^{-1}(0) = \sin^{-1}(1) - 0 = \frac{\pi}{2} \]
This is valid.

2. For \( x = \frac{1}{2} \):
\[ \sin^{-1}(1 - \frac{1}{2}) - 2\sin^{-1}(\frac{1}{2}) = \sin^{-1}(\frac{1}{2}) - 2\left(\frac{\pi}{6}\right) = \frac{\pi}{6} - \frac{\pi}{3} = -\frac{\pi}{6} \]
This is not valid.

### Conclusion
The only valid solution is:
\[ \boxed{0} \]