The value of `sin{cot^(-1)[cos(cot^(-1)((1)/(x)))]}` is equal to `(x gt0)`

Options

A. `sqrt((1+x^(2))/(2+x^(2)))`
B. `sqrt((1-x^(2))/(2+x^(2)))`
C. `sqrt((1+x^(2))/(2-x^(2)))`
D. `sqrt((2+x^(2))/(1+x^(2)))`

Answer

To solve the problem, we need to find the value of \( \sin\left(\cot^{-1}\left[\cos\left(\cot^{-1}\left(\frac{1}{x}\right)\right)\right]\right) \) for \( x > 0 \).

### Step-by-Step Solution:

1. **Let \( \alpha = \cot^{-1}\left(\frac{1}{x}\right) \)**:
- By definition, \( \cot(\alpha) = \frac{1}{x} \).
- This implies \( x = \frac{1}{\cot(\alpha)} = \tan(\alpha) \).

**Hint**: Remember that \( \cot^{-1}(y) \) gives an angle whose cotangent is \( y \).

2. **Find \( \cos(\alpha) \)**:
- In a right triangle where \( \tan(\alpha) = x \), we can represent the sides as follows:
- Opposite side = \( x \)
- Adjacent side = \( 1 \)
- Using the Pythagorean theorem, the hypotenuse \( h \) is given by:
\[ h = \sqrt{x^2 + 1^2} = \sqrt{1 + x^2} \]
- Therefore, \( \cos(\alpha) = \frac{\text{Adjacent}}{\text{Hypotenuse}} = \frac{1}{\sqrt{1 + x^2}} \).

**Hint**: Use the triangle relationships to find cosine based on tangent.

3. **Now find \( \cot^{-1}(\cos(\alpha)) \)**:
- We have \( \cos(\alpha) = \frac{1}{\sqrt{1 + x^2}} \).
- Let \( \beta = \cot^{-1}\left(\frac{1}{\sqrt{1 + x^2}}\right) \).
- This means \( \cot(\beta) = \frac{1}{\sqrt{1 + x^2}} \).

**Hint**: Recognize that \( \cot^{-1}(y) \) gives an angle whose cotangent is \( y \).

4. **Find \( \sin(\beta) \)**:
- In a right triangle where \( \cot(\beta) = \frac{1}{\sqrt{1 + x^2}} \):
- Opposite side = \( 1 \)
- Adjacent side = \( \sqrt{1 + x^2} \)
- The hypotenuse \( h \) is given by:
\[ h = \sqrt{1^2 + \left(\sqrt{1 + x^2}\right)^2} = \sqrt{1 + (1 + x^2)} = \sqrt{2 + x^2} \]
- Therefore, \( \sin(\beta) = \frac{\text{Opposite}}{\text{Hypotenuse}} = \frac{1}{\sqrt{2 + x^2}} \).

**Hint**: Again, use the triangle relationships to find sine based on cotangent.

5. **Final Result**:
- Thus, we have:
\[ \sin\left(\cot^{-1}\left[\cos\left(\cot^{-1}\left(\frac{1}{x}\right)\right)\right]\right) = \frac{1}{\sqrt{2 + x^2}} \]

### Conclusion:
The value of \( \sin\left(\cot^{-1}\left[\cos\left(\cot^{-1}\left(\frac{1}{x}\right)\right)\right]\right) \) is \( \frac{1}{\sqrt{2 + x^2}} \) for \( x > 0 \).