If `sin^(-1)a+sin^(-1)b+sin^(-1)c=pi,` then the value of `asqrt((1-a^2))+bsqrt((1-b^2))+sqrt((1-c^2))` will be `2a b c` (b) `a b c` (c) `1/2a b c` (d) `1/3a b c`

Options

A. `2 abc`
B. `abc`
C. `(1)/(2) abc`
D. `(1)/(3) abc`

Explanation

To solve the problem, we start with the given equation:

\[ \sin^{-1} a + \sin^{-1} b + \sin^{-1} c = \pi \]

### Step 1: Express the angles
Let:
\[ \sin^{-1} a = A, \quad \sin^{-1} b = B, \quad \sin^{-1} c = C \]
Thus, we have:
\[ A + B + C = \pi \]

### Step 2: Use the sine function
From the definitions of inverse sine, we can express \(a\), \(b\), and \(c\) as:
\[ a = \sin A, \quad b = \sin B, \quad c = \sin C \]

### Step 3: Recognize the triangle property
Since \(A + B + C = \pi\), \(A\), \(B\), and \(C\) can be considered as the angles of a triangle. We can apply the sine rule for the angles of a triangle, which states:
\[ \sin A + \sin B + \sin C = 4 \sin A \sin B \sin C \]

### Step 4: Substitute the sine values
Using the sine values we defined:
\[ \sin A = a, \quad \sin B = b, \quad \sin C = c \]
We can rewrite the equation as:
\[ a + b + c = 4abc \]

### Step 5: Find the required expression
We need to evaluate:
\[ a \sqrt{1 - a^2} + b \sqrt{1 - b^2} + c \sqrt{1 - c^2} \]
Using the identity \(\sqrt{1 - \sin^2 x} = \cos x\), we can rewrite the expression as:
\[ a \cos A + b \cos B + c \cos C \]

### Step 6: Express cosines in terms of sine
We know:
\[ \cos A = \sqrt{1 - a^2}, \quad \cos B = \sqrt{1 - b^2}, \quad \cos C = \sqrt{1 - c^2} \]
Thus, we can express the required sum as:
\[ a \sqrt{1 - a^2} + b \sqrt{1 - b^2} + c \sqrt{1 - c^2} = a \cos A + b \cos B + c \cos C \]

### Step 7: Use the triangle property
From the triangle properties, we know:
\[ \sin A \cos A + \sin B \cos B + \sin C \cos C = \frac{1}{2} \sin 2A + \frac{1}{2} \sin 2B + \frac{1}{2} \sin 2C \]
This can be simplified using the sine of angles in a triangle.

### Step 8: Final result
Using the earlier derived relation, we find:
\[ a \sqrt{1 - a^2} + b \sqrt{1 - b^2} + c \sqrt{1 - c^2} = 2abc \]

Thus, the value of \(a \sqrt{1 - a^2} + b \sqrt{1 - b^2} + c \sqrt{1 - c^2}\) is:

\[ \boxed{2abc} \]