If `sin^(-1)x+sin^(-1)y+sin^(-1)z=(3pi)/2`, Then `x^2013+y^2013+z^2013-9/(x^2013+y^2013+z^2013)` is equal to

Options

A. 1
B. 0
C. -1
D. -3

Answer

To solve the equation \( \sin^{-1} x + \sin^{-1} y + \sin^{-1} z = \frac{3\pi}{2} \), we can follow these steps:

### Step 1: Analyze the equation
The function \( \sin^{-1} t \) (the inverse sine function) has a range of \( \left[-\frac{\pi}{2}, \frac{\pi}{2}\right] \). Therefore, the maximum value of \( \sin^{-1} x + \sin^{-1} y + \sin^{-1} z \) occurs when each term is at its maximum, which is \( \frac{\pi}{2} \).

### Step 2: Set each term equal to its maximum
Since the sum of three maximum values \( \frac{\pi}{2} + \frac{\pi}{2} + \frac{\pi}{2} = \frac{3\pi}{2} \), we can conclude that:
\[ \sin^{-1} x = \sin^{-1} y = \sin^{-1} z = \frac{\pi}{2} \]

### Step 3: Solve for \( x, y, z \)
From \( \sin^{-1} x = \frac{\pi}{2} \), we find:
\[ x = \sin\left(\frac{\pi}{2}\right) = 1 \]
Similarly, we have:
\[ y = 1 \quad \text{and} \quad z = 1 \]

### Step 4: Substitute values into the expression
Now we need to evaluate the expression:
\[ x^{2013} + y^{2013} + z^{2013} - \frac{9}{x^{2013} + y^{2013} + z^{2013}} \]
Substituting \( x = 1, y = 1, z = 1 \):
\[ x^{2013} + y^{2013} + z^{2013} = 1^{2013} + 1^{2013} + 1^{2013} = 1 + 1 + 1 = 3 \]

### Step 5: Calculate the final expression
Now substituting back into the expression:
\[ 3 - \frac{9}{3} = 3 - 3 = 0 \]

Thus, the final answer is:
\[ \boxed{0} \]