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

Options

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

Answer

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

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

### Step 1: Analyze the equation

The maximum value of \(\sin^{-1} t\) for any real number \(t\) is \(\frac{\pi}{2}\). This implies that for the sum of three inverse sine functions to equal \(\frac{3\pi}{2}\), each term must equal \(\frac{\pi}{2}\):

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

### Step 2: Solve for \(x\), \(y\), and \(z\)

From \(\sin^{-1}x = \frac{\pi}{2}\), we have:

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

Similarly, for \(y\) and \(z\):

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

Thus, we find:

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

### Step 3: Calculate \(x^{100} + y^{100} + z^{100}\)

Now we compute:

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

### Step 4: Calculate \(x^{100} + y^{100} + z^{101}\)

Next, we compute:

\[ x^{100} + y^{100} + z^{101} = 1^{100} + 1^{100} + 1^{101} = 1 + 1 + 1 = 3 \]

### Step 5: Substitute into the expression

Now we substitute these values into the expression we need to evaluate:

\[ x^{100} + y^{100} + z^{100} - \frac{9}{x^{100} + y^{100} + z^{101}} = 3 - \frac{9}{3} \]

Calculating the fraction:

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

Thus, we have:

\[ 3 - 3 = 0 \]

### Final Answer

The final value is:

\[ \boxed{0} \]