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 want to find the value of:

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

### Step 1: Identify the maximum values of the inverse trigonometric functions

- 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 \(\pi\).
- The maximum value of \(\csc^{-1} d\) is \(\frac{\pi}{2}\).

### Step 2: Substitute maximum values into the equation

Substituting the maximum values into the equation:

\[ \left(\frac{\pi}{2}\right)^2 + \pi^2 + \pi^2 + \left(\frac{\pi}{2}\right)^2 = \frac{5\pi^2}{2} \]

Calculating each term:

- \(\left(\frac{\pi}{2}\right)^2 = \frac{\pi^2}{4}\)
- \(\pi^2\) (for both \(\cos^{-1} b\) and \(\sec^{-1} c\))

So we have:

\[ \frac{\pi^2}{4} + \pi^2 + \pi^2 + \frac{\pi^2}{4} = \frac{\pi^2}{4} + \frac{4\pi^2}{4} + \frac{\pi^2}{4} = \frac{6\pi^2}{4} = \frac{3\pi^2}{2} \]

This does not equal \(\frac{5\pi^2}{2}\), so we need to adjust our values.

### Step 3: Set values based on the equation

From the equation, we can set:

Let:
- \(\sin^{-1} a = \frac{\pi}{2}\)
- \(\cos^{-1} b = \pi\)
- \(\sec^{-1} c = \pi\)
- \(\csc^{-1} d = \frac{\pi}{2}\)

This gives us:

\[ \left(\frac{\pi}{2}\right)^2 + \pi^2 + \pi^2 + \left(\frac{\pi}{2}\right)^2 = \frac{5\pi^2}{2} \]

### Step 4: Calculate the desired expression

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

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

Substituting the values:

\[ \left(\frac{\pi}{2}\right)^2 - \pi^2 + \pi^2 - \left(\frac{\pi}{2}\right)^2 \]

Calculating this gives:

\[ \frac{\pi^2}{4} - \pi^2 + \pi^2 - \frac{\pi^2}{4} \]

The \(-\pi^2\) and \(+\pi^2\) cancel out:

\[ \frac{\pi^2}{4} - \frac{\pi^2}{4} = 0 \]

### Final Answer

Thus, the value of the expression is:

\[ \boxed{0} \]