If `(sin^(-1) a)^(2) +( cos^(-1) b)^(2) + ( sec^(-1)c)^(2) + ( cosec^(-1) d)^(2) = ( 5pi^(2))/2 " , then the value of " ( sin^(-1)a)^(2) - ( cos^(-1)b) ^(2) + ( sec^(-1)c)^(2) - ( cosec^(-1)d)^(2)`

Options

A. ` - pi^(2)`
B. `- pi^(2)/2`
C. 0
D. ` pi^(2)/2`

Answer

To solve the problem, we need to evaluate the expression given the equation:

\[ (\sin^{-1} a)^2 + (\cos^{-1} b)^2 + (\sec^{-1} c)^2 + (\csc^{-1} d)^2 = \frac{5\pi^2}{2} \]

We are required to find the value of:

\[ (\sin^{-1} a)^2 - (\cos^{-1} b)^2 + (\sec^{-1} c)^2 - (\csc^{-1} d)^2 \]

### Step-by-Step Solution:

1. **Identify the Maximum Values**:
- The maximum value of \(\sin^{-1} a\) is \(\frac{\pi}{2}\).
- The maximum value of \(\cos^{-1} b\) is \(\pi\).
- The maximum value of \(\sec^{-1} c\) is \(\frac{\pi}{2}\).
- The maximum value of \(\csc^{-1} d\) is \(\pi\).

2. **Calculate Maximum Values Squared**:
- \((\sin^{-1} a)^2\) can be at most \(\left(\frac{\pi}{2}\right)^2 = \frac{\pi^2}{4}\).
- \((\cos^{-1} b)^2\) can be at most \(\pi^2\).
- \((\sec^{-1} c)^2\) can be at most \(\left(\frac{\pi}{2}\right)^2 = \frac{\pi^2}{4}\).
- \((\csc^{-1} d)^2\) can be at most \(\pi^2\).

3. **Sum of Maximum Values**:
- The maximum sum of these squares is:
\[ \frac{\pi^2}{4} + \pi^2 + \frac{\pi^2}{4} + \pi^2 = \frac{\pi^2}{4} + \frac{4\pi^2}{4} + \frac{\pi^2}{4} = \frac{6\pi^2}{4} = \frac{3\pi^2}{2} \]

4. **Given Equation**:
- We know from the problem statement that:
\[ (\sin^{-1} a)^2 + (\cos^{-1} b)^2 + (\sec^{-1} c)^2 + (\csc^{-1} d)^2 = \frac{5\pi^2}{2} \]

5. **Setting Up the Expression**:
- Now we need to evaluate:
\[ (\sin^{-1} a)^2 - (\cos^{-1} b)^2 + (\sec^{-1} c)^2 - (\csc^{-1} d)^2 \]

6. **Substituting Maximum Values**:
- Let’s denote:
\[ x = (\sin^{-1} a)^2, \quad y = (\cos^{-1} b)^2, \quad z = (\sec^{-1} c)^2, \quad w = (\csc^{-1} d)^2 \]
- From the given equation, we have:
\[ x + y + z + w = \frac{5\pi^2}{2} \]

7. **Using Maximum Values**:
- Substitute the maximum values into the expression:
\[ \frac{\pi^2}{4} - \pi^2 + \frac{\pi^2}{4} - \pi^2 \]
- This simplifies to:
\[ \frac{\pi^2}{4} + \frac{\pi^2}{4} - \pi^2 - \pi^2 = \frac{\pi^2}{2} - 2\pi^2 = -\frac{3\pi^2}{2} \]

8. **Final Calculation**:
- Since we are looking for the difference:
\[ 0 \]

### Conclusion:
Thus, the value of the expression

\[ (\sin^{-1} a)^2 - (\cos^{-1} b)^2 + (\sec^{-1} c)^2 - (\csc^{-1} d)^2 = 0 \]