If `(sin^(-1) x)^(2) - (cos^(-1) x)^(2) = a pi^(2)` then find the range of a

Options

A. `[-(3)/(4), (1)/(4)]`
B. `[-(3)/(4), (3)/(4)]`
C. `[-1, 1]`
D. `[-1, (3)/(4)]`

Explanation

To solve the equation \((\sin^{-1} x)^2 - (\cos^{-1} x)^2 = a \pi^2\), we can follow these steps:

### Step 1: Use the identity for inverse trigonometric functions
We know that:
\[ \sin^{-1} x + \cos^{-1} x = \frac{\pi}{2} \]
Let \(y = \sin^{-1} x\). Then, we can express \(\cos^{-1} x\) as:
\[ \cos^{-1} x = \frac{\pi}{2} - y \]

### Step 2: Substitute into the equation
Substituting \(\cos^{-1} x\) into the original equation gives:
\[ y^2 - \left(\frac{\pi}{2} - y\right)^2 = a \pi^2 \]

### Step 3: Expand the expression
Now, we expand \(\left(\frac{\pi}{2} - y\right)^2\):
\[ \left(\frac{\pi}{2} - y\right)^2 = \left(\frac{\pi^2}{4} - \pi y + y^2\right) \]
Thus, the equation becomes:
\[ y^2 - \left(\frac{\pi^2}{4} - \pi y + y^2\right) = a \pi^2 \]

### Step 4: Simplify the equation
Now, simplify the left-hand side:
\[ y^2 - \frac{\pi^2}{4} + \pi y - y^2 = a \pi^2 \]
This simplifies to:
\[ \pi y - \frac{\pi^2}{4} = a \pi^2 \]

### Step 5: Solve for \(y\)
Rearranging gives:
\[ \pi y = a \pi^2 + \frac{\pi^2}{4} \]
Dividing through by \(\pi\) (assuming \(\pi \neq 0\)):
\[ y = a \pi + \frac{\pi}{4} \]

### Step 6: Find the range of \(y\)
Since \(y = \sin^{-1} x\), we know that:
\[ -\frac{\pi}{2} \leq y \leq \frac{\pi}{2} \]
Substituting our expression for \(y\):
\[ -\frac{\pi}{2} \leq a \pi + \frac{\pi}{4} \leq \frac{\pi}{2} \]

### Step 7: Solve the inequalities
1. For the left inequality:
\[ a \pi + \frac{\pi}{4} \geq -\frac{\pi}{2} \]
Simplifying gives:
\[ a \pi \geq -\frac{\pi}{2} - \frac{\pi}{4} = -\frac{3\pi}{4} \]
Dividing by \(\pi\) (assuming \(\pi > 0\)):
\[ a \geq -\frac{3}{4} \]

2. For the right inequality:
\[ a \pi + \frac{\pi}{4} \leq \frac{\pi}{2} \]
Simplifying gives:
\[ a \pi \leq \frac{\pi}{2} - \frac{\pi}{4} = \frac{\pi}{4} \]
Dividing by \(\pi\):
\[ a \leq \frac{1}{4} \]

### Step 8: Combine the results
Combining both inequalities, we have:
\[ -\frac{3}{4} \leq a \leq \frac{1}{4} \]

### Final Answer
The range of \(a\) is:
\[ \boxed{[-\frac{3}{4}, \frac{1}{4}]} \]