If ` x + 1/x = 2`, the principal value of `sin^(-1) x` is

Options

A. `pi/4`
B. `pi/2`
C. `pi`
D. `(3pi)/2`

Answer

To solve the problem, we need to find the principal value of \( \sin^{-1}(x) \) given that \( x + \frac{1}{x} = 2 \).

### Step-by-Step Solution:

1. **Start with the given equation:**
\[ x + \frac{1}{x} = 2 \]

2. **Multiply both sides by \( x \) to eliminate the fraction:**
\[ x^2 + 1 = 2x \]

3. **Rearrange the equation:**
\[ x^2 - 2x + 1 = 0 \]

4. **Factor the quadratic equation:**
\[ (x - 1)^2 = 0 \]

5. **Solve for \( x \):**
\[ x - 1 = 0 \implies x = 1 \]

6. **Now, find the principal value of \( \sin^{-1}(x) \):**
\[ \sin^{-1}(1) \]

7. **Recall that \( \sin(\frac{\pi}{2}) = 1 \):**
\[ \sin^{-1}(1) = \frac{\pi}{2} \]

8. **Conclusion:**
The principal value of \( \sin^{-1}(x) \) is:
\[ \frac{\pi}{2} \]

### Final Answer:
\[ \frac{\pi}{2} \]

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