If `sin^(-1)x+sin^(-1)y+sin^(-1)z=(3pi)/(2)` the value of `x^(100)+y^(100)+z^(100)-(9)/(x^(101)+y^(101)+z^(101))`is

Options

A. 0
B. 1
C. 2
D. 3

Answer

To solve the problem, we start with the equation given:

\[ \sin^{-1}x + \sin^{-1}y + \sin^{-1}z = \frac{3\pi}{2} \]

### Step 1: Analyze the equation
The range of the inverse sine function, \(\sin^{-1}\), is from \(-\frac{\pi}{2}\) to \(\frac{\pi}{2}\). Therefore, the maximum sum of three inverse sine functions can be at most \(3 \cdot \frac{\pi}{2} = \frac{3\pi}{2}\). This means that for the equation to hold, each of \(\sin^{-1}x\), \(\sin^{-1}y\), and \(\sin^{-1}z\) must be equal to \(\frac{\pi}{2}\).

### Step 2: Find values of \(x\), \(y\), and \(z\)
Since \(\sin^{-1}x = \frac{\pi}{2}\), we have:

\[ x = \sin\left(\frac{\pi}{2}\right) = 1 \]

Similarly, we find:

\[ y = \sin\left(\frac{\pi}{2}\right) = 1 \]
\[ z = \sin\left(\frac{\pi}{2}\right) = 1 \]

Thus, we have:

\[ x = 1, \quad y = 1, \quad z = 1 \]

### Step 3: Substitute values into the expression
Now we substitute \(x\), \(y\), and \(z\) into the expression we need to evaluate:

\[ x^{100} + y^{100} + z^{100} - \frac{9}{x^{101} + y^{101} + z^{101}} \]

Calculating each term:

\[ x^{100} = 1^{100} = 1 \]
\[ y^{100} = 1^{100} = 1 \]
\[ z^{100} = 1^{100} = 1 \]

So:

\[ x^{100} + y^{100} + z^{100} = 1 + 1 + 1 = 3 \]

Next, we calculate \(x^{101} + y^{101} + z^{101}\):

\[ x^{101} = 1^{101} = 1 \]
\[ y^{101} = 1^{101} = 1 \]
\[ z^{101} = 1^{101} = 1 \]

Thus:

\[ x^{101} + y^{101} + z^{101} = 1 + 1 + 1 = 3 \]

### Step 4: Substitute back into the expression
Now we can substitute these values back into the expression:

\[ 3 - \frac{9}{3} = 3 - 3 = 0 \]

### Final Answer
The value of the expression is:

\[ \boxed{0} \]