If `(sin^(-1)x+sin^(-1)w)(sin^(-1)y+sin^(-1)z)=pi^2,` then `D=|x^(N_1)y^(N_3)z^(N_3)w^(N_4)|(N_1,N_2,N_3,N_4 in N)` has a maximum value of 2 has a maximum value of 0 16 different D are possible has a minimum value of `-2`

Options

A. has a maximum value of 2
B. has a minimum value of 0
C. 16 different D are possible
D. has a minimum value of `-2`

Explanation

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

\[ (\sin^{-1} x + \sin^{-1} w)(\sin^{-1} y + \sin^{-1} z) = \pi^2 \]

### Step 1: Analyze the equation
The maximum value of \(\sin^{-1} x\) and \(\sin^{-1} w\) occurs when \(x = 1\) and \(w = 1\). In that case, we have:

\[ \sin^{-1} x + \sin^{-1} w = \sin^{-1}(1) + \sin^{-1}(1) = \frac{\pi}{2} + \frac{\pi}{2} = \pi \]

Similarly, for \(\sin^{-1} y + \sin^{-1} z\):

\[ \sin^{-1} y + \sin^{-1} z = \pi \]

This means that both pairs must equal \(\pi\) for the product to equal \(\pi^2\).

### Step 2: Consider the possible values
From the above, we can conclude that:

1. \(x = 1, w = 1\) or \(x = -1, w = -1\)
2. \(y = 1, z = 1\) or \(y = -1, z = -1\)

Thus, \(x, w, y, z\) can take values of either \(1\) or \(-1\).

### Step 3: Calculate \(D\)
Given \(D = |x^{N_1} y^{N_2} z^{N_3} w^{N_4}|\), we can evaluate the maximum value of \(D\).

- If \(N_1, N_2, N_3, N_4\) are all \(1\), then:

\[ D = |1^{N_1} \cdot 1^{N_2} \cdot 1^{N_3} \cdot 1^{N_4}| = |1| = 1 \]

- If \(N_1, N_2, N_3, N_4\) are all \(-1\), then:

\[ D = |-1^{N_1} \cdot -1^{N_2} \cdot -1^{N_3} \cdot -1^{N_4}| = |-1| = 1 \]

### Step 4: Count combinations
Each of \(x, w, y, z\) can independently be either \(1\) or \(-1\). Therefore, there are \(2^4 = 16\) different combinations of values for \(D\).

### Conclusion
Thus, the maximum value of \(D\) is \(1\) and there are \(16\) different possible values for \(D\).

### Final Answer
The correct option is: **16 different \(D\) are possible.**