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 problem, we need to find the range of \( a \) given the equation:

\[ (\sin^{-1} x)^2 - (\cos^{-1} x)^2 = a \pi^2 \]

### Step 1: Use the identity for the difference of squares
We can rewrite the left-hand side using the identity \( a^2 - b^2 = (a - b)(a + b) \):

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

### Step 2: Use the known identity for inverse trigonometric functions
We know that:

\[ \sin^{-1} x + \cos^{-1} x = \frac{\pi}{2} \]

Substituting this into our equation gives:

\[ (\sin^{-1} x - \cos^{-1} x) \cdot \frac{\pi}{2} = a \pi^2 \]

### Step 3: Simplify the equation
Now we can simplify this equation:

\[ \sin^{-1} x - \cos^{-1} x = a \pi \cdot 2 \]

### Step 4: Express \(\cos^{-1} x\) in terms of \(\sin^{-1} x\)
We can express \(\cos^{-1} x\) as:

\[ \cos^{-1} x = \frac{\pi}{2} - \sin^{-1} x \]

Substituting this into our equation gives:

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

### Step 5: Combine like terms
This simplifies to:

\[ 2 \sin^{-1} x - \frac{\pi}{2} = 2a \pi \]

### Step 6: Rearrange the equation
Rearranging gives:

\[ 2 \sin^{-1} x = 2a \pi + \frac{\pi}{2} \]

### Step 7: Divide by 2
Dividing the entire equation by 2 gives:

\[ \sin^{-1} x = a \pi + \frac{\pi}{4} \]

### Step 8: Determine the range of \(\sin^{-1} x\)
The range of \(\sin^{-1} x\) is:

\[ -\frac{\pi}{2} \leq \sin^{-1} x \leq \frac{\pi}{2} \]

### Step 9: Set up inequalities
Now we can set up inequalities based on the expression we derived:

1. From the lower bound:
\[ a \pi + \frac{\pi}{4} \geq -\frac{\pi}{2} \]
Simplifying this gives:
\[ a \pi \geq -\frac{\pi}{2} - \frac{\pi}{4} = -\frac{3\pi}{4} \]
Dividing by \(\pi\) (and reversing the inequality since \(\pi > 0\)):
\[ a \geq -\frac{3}{4} \]

2. From the upper bound:
\[ a \pi + \frac{\pi}{4} \leq \frac{\pi}{2} \]
Simplifying this gives:
\[ a \pi \leq \frac{\pi}{2} - \frac{\pi}{4} = \frac{\pi}{4} \]
Dividing by \(\pi\):
\[ a \leq \frac{1}{4} \]

### Final Step: Combine the inequalities
Combining these results, we find:

\[ -\frac{3}{4} \leq a \leq \frac{1}{4} \]

Thus, the range of \( a \) is:

\[ \boxed{\left[-\frac{3}{4}, \frac{1}{4}\right]} \]