If `cos^-1p+cos^-1q+cos^-1r=pi(0 <= p,q,r <= 1).` then the value of `p^2+q^2+r^2+2pqr` is

Options

A. 0
B. 1
C. 2
D. none of these

Answer

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

\[ \cos^{-1} p + \cos^{-1} q + \cos^{-1} r = \pi \]

### Step 1: Rearranging the Equation
We can rearrange the equation to isolate one of the terms:

\[ \cos^{-1} p + \cos^{-1} q = \pi - \cos^{-1} r \]

### Step 2: Using the Cosine Addition Formula
Using the cosine addition formula, we know that:

\[ \cos^{-1} p + \cos^{-1} q = \cos^{-1}(pq - \sqrt{(1 - p^2)(1 - q^2)}) \]

Thus, we can write:

\[ pq - \sqrt{(1 - p^2)(1 - q^2)} = \cos(\pi - \cos^{-1} r) \]

### Step 3: Simplifying the Right Side
The cosine of \(\pi - x\) is \(-\cos x\), therefore:

\[ pq - \sqrt{(1 - p^2)(1 - q^2)} = -r \]

### Step 4: Rearranging the Equation
Rearranging gives us:

\[ pq + r = \sqrt{(1 - p^2)(1 - q^2)} \]

### Step 5: Squaring Both Sides
Now we square both sides:

\[ (pq + r)^2 = (1 - p^2)(1 - q^2) \]

### Step 6: Expanding Both Sides
Expanding both sides yields:

\[ p^2q^2 + 2pqr + r^2 = 1 - p^2 - q^2 + p^2q^2 \]

### Step 7: Simplifying the Equation
We can simplify this by cancelling \(p^2q^2\) from both sides:

\[ 2pqr + r^2 = 1 - p^2 - q^2 \]

### Step 8: Rearranging to Find the Desired Expression
Rearranging gives:

\[ p^2 + q^2 + r^2 + 2pqr = 1 \]

### Conclusion
Thus, we find that:

\[ p^2 + q^2 + r^2 + 2pqr = 1 \]

### Final Answer
The value of \(p^2 + q^2 + r^2 + 2pqr\) is:

\[ \boxed{1} \]