The sum of all possible values of x satisfying the equation `sin^(-1)(3x-4x^(3))+cos^(-1)(4x^(3)-3x)=(pi)/(2)` is

Options

A. `-2`
B. `-1`
C. 1
D. 0

Explanation

To solve the equation \( \sin^{-1}(3x - 4x^3) + \cos^{-1}(4x^3 - 3x) = \frac{\pi}{2} \), we can use the property that \( \sin^{-1}(y) + \cos^{-1}(y) = \frac{\pi}{2} \) for any \( y \). This means that we can rewrite the equation as follows:

### Step 1: Rewrite the equation
Using the identity, we can rewrite the equation:
\[ \sin^{-1}(3x - 4x^3) = \frac{\pi}{2} - \cos^{-1}(4x^3 - 3x) \]
This implies:
\[ \sin^{-1}(3x - 4x^3) = \sin^{-1}(- (3x - 4x^3)) \]

### Step 2: Set the arguments equal
Since the inverse sine function is one-to-one, we can set the arguments equal:
\[ 3x - 4x^3 = - (3x - 4x^3) \]

### Step 3: Simplify the equation
This simplifies to:
\[ 3x - 4x^3 = -3x + 4x^3 \]
Adding \( 3x \) and \( 4x^3 \) to both sides gives:
\[ 3x + 3x = 4x^3 + 4x^3 \]
This simplifies to:
\[ 6x = 8x^3 \]

### Step 4: Rearrange the equation
Rearranging gives:
\[ 8x^3 - 6x = 0 \]

### Step 5: Factor the equation
Factoring out \( 2x \) gives:
\[ 2x(4x^2 - 3) = 0 \]

### Step 6: Solve for x
Setting each factor to zero gives:
1. \( 2x = 0 \) which implies \( x = 0 \)
2. \( 4x^2 - 3 = 0 \) which implies \( 4x^2 = 3 \) or \( x^2 = \frac{3}{4} \), thus \( x = \pm \frac{\sqrt{3}}{2} \)

### Step 7: List all possible values of x
The possible values of \( x \) are:
- \( x = 0 \)
- \( x = \frac{\sqrt{3}}{2} \)
- \( x = -\frac{\sqrt{3}}{2} \)

### Step 8: Find the sum of all possible values of x
Now, we sum all the possible values:
\[ 0 + \frac{\sqrt{3}}{2} - \frac{\sqrt{3}}{2} = 0 \]

### Final Answer
The sum of all possible values of \( x \) satisfying the equation is:
\[ \boxed{0} \]