Maximum value of function `f(x)=(sin^(-1)(sinx)^(2)-sin^(-1)(sinx)` is:

Options

A. `(pi)/(4)[pi+2]`
B. `(pi)/(4)[pi-2]`
C. `(pi)/(2)[pi+2]`
D. `(pi)/(2)[pi-2]`

Explanation

To find the maximum value of the function \( f(x) = \sin^{-1}(\sin x)^2 - \sin^{-1}(\sin x) \), we can follow these steps:

### Step 1: Substitute \( t \)
Let \( t = \sin^{-1}(\sin x) \). Then, we can rewrite the function as:
\[ f(x) = t^2 - t \]

### Step 2: Complete the square
To maximize \( f(x) \), we can complete the square for the expression \( t^2 - t \):
\[ f(x) = t^2 - t = \left(t - \frac{1}{2}\right)^2 - \frac{1}{4} \]

### Step 3: Determine the range of \( t \)
Next, we need to find the maximum and minimum values of \( t = \sin^{-1}(\sin x) \). The function \( \sin^{-1}(\sin x) \) takes values in the interval:
\[ \left[-\frac{\pi}{2}, \frac{\pi}{2}\right] \]

### Step 4: Find the maximum value of \( t \)
The maximum value of \( t \) is \( \frac{\pi}{2} \) and the minimum value is \( -\frac{\pi}{2} \).

### Step 5: Substitute the maximum value of \( t \)
To find the maximum value of \( f(x) \), substitute \( t = \frac{\pi}{2} \) into the completed square expression:
\[ f(x) = \left(\frac{\pi}{2} - \frac{1}{2}\right)^2 - \frac{1}{4} \]

### Step 6: Simplify the expression
Calculating the expression:
\[ f(x) = \left(\frac{\pi}{2} - \frac{1}{2}\right)^2 - \frac{1}{4} = \left(\frac{\pi - 1}{2}\right)^2 - \frac{1}{4} \]
\[ = \frac{(\pi - 1)^2}{4} - \frac{1}{4} \]
\[ = \frac{(\pi - 1)^2 - 1}{4} \]

### Step 7: Final expression
Thus, the maximum value of the function \( f(x) \) is:
\[ f(x) = \frac{\pi^2 - 2\pi}{4} \]

### Conclusion
The maximum value of the function \( f(x) = \sin^{-1}(\sin x)^2 - \sin^{-1}(\sin x) \) is:
\[ \frac{\pi^2 - 2\pi}{4} \]