The value of `x` satisfying the equation
`(sin^(-1)x)^(3)-(cos^(-1)x)^(3)+(sin^(-1)x)(cos^(-1)x)(sin^(-1)x-cos^(-1)x)=(pi^(3))/(16)` is : (a) `"cos"(pi)/(5)` (b) `"cos"(pi)/(4)` (c) `"cos"(pi)/(8)` (d) `"cos"(pi)/(12)`

Options

A. `"cos"(pi)/(5)`
B. `"cos"(pi)/(4)`
C. `"cos"(pi)/(8)`
D. `"cos"(pi)/(12)`

Answer

To solve the equation

\[ (\sin^{-1} x)^3 - (\cos^{-1} x)^3 + (\sin^{-1} x)(\cos^{-1} x)(\sin^{-1} x - \cos^{-1} x) = \frac{\pi^3}{16}, \]

we will follow these steps:

### Step 1: Define Variables
Let \( A = \sin^{-1} x \) and \( B = \cos^{-1} x \).

### Step 2: Rewrite the Equation
The equation can be rewritten using \( A \) and \( B \):

\[ A^3 - B^3 + AB(A - B) = \frac{\pi^3}{16}. \]

### Step 3: Apply the Identity for Difference of Cubes
Recall the identity for the difference of cubes:

\[ A^3 - B^3 = (A - B)(A^2 + AB + B^2). \]

Using this identity, we can rewrite the equation:

\[ (A - B)(A^2 + AB + B^2) + AB(A - B) = \frac{\pi^3}{16}. \]

### Step 4: Factor Out \( A - B \)
Factor out \( A - B \):

\[ (A - B)(A^2 + AB + B^2 + AB) = \frac{\pi^3}{16}. \]

This simplifies to:

\[ (A - B)(A^2 + 2AB + B^2) = \frac{\pi^3}{16}. \]

### Step 5: Recognize the Perfect Square
Notice that \( A^2 + 2AB + B^2 = (A + B)^2 \):

\[ (A - B)(A + B)^2 = \frac{\pi^3}{16}. \]

### Step 6: Use the Identity for \( A + B \)
We know that:

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

Thus, we can substitute:

\[ (A - B)\left(\frac{\pi}{2}\right)^2 = \frac{\pi^3}{16}. \]

### Step 7: Simplify the Equation
Substituting \( \left(\frac{\pi}{2}\right)^2 \):

\[ (A - B)\left(\frac{\pi^2}{4}\right) = \frac{\pi^3}{16}. \]

### Step 8: Solve for \( A - B \)
Now, we can solve for \( A - B \):

\[ A - B = \frac{\pi^3}{16} \cdot \frac{4}{\pi^2} = \frac{\pi}{4}. \]

### Step 9: Solve for \( B \)
We have two equations now:

1. \( A + B = \frac{\pi}{2} \)
2. \( A - B = \frac{\pi}{4} \)

Adding these two equations:

\[ 2A = \frac{\pi}{2} + \frac{\pi}{4} = \frac{2\pi + \pi}{4} = \frac{3\pi}{4} \implies A = \frac{3\pi}{8}. \]

Now, substituting \( A \) back to find \( B \):

\[ B = \frac{\pi}{2} - A = \frac{\pi}{2} - \frac{3\pi}{8} = \frac{4\pi - 3\pi}{8} = \frac{\pi}{8}. \]

### Step 10: Find \( x \)
Since \( B = \cos^{-1} x \), we have:

\[ \cos^{-1} x = \frac{\pi}{8} \implies x = \cos\left(\frac{\pi}{8}\right). \]

### Conclusion
Thus, the value of \( x \) satisfying the equation is:

\[ \boxed{\cos\left(\frac{\pi}{8}\right)}. \]