If `cos(cot^(-1)((1)/(2)))=cot(cos^(-1)x)`, then a value of x is

Options

A. `(1)/(sqrt(6))`
B. `(-1)/(sqrt(2))`
C. `(2)/(sqrt(6))`
D. `(-2)/(sqrt(6))`

Answer

To solve the equation \( \cos(\cot^{-1}(\frac{1}{2})) = \cot(\cos^{-1}(x)) \), we will follow these steps:

### Step 1: Solve the left-hand side \( \cos(\cot^{-1}(\frac{1}{2})) \)

1. **Construct a right triangle** where the angle \( \theta = \cot^{-1}(\frac{1}{2}) \).
- In this triangle, the adjacent side can be \( 1 \) and the opposite side can be \( 2 \) (since \( \cot \theta = \frac{\text{adjacent}}{\text{opposite}} = \frac{1}{2} \)).
2. **Find the hypotenuse** using the Pythagorean theorem:
\[ \text{hypotenuse} = \sqrt{(1^2 + 2^2)} = \sqrt{5} \]
3. **Calculate \( \cos \theta \)**:
\[ \cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{1}{\sqrt{5}} \]

### Step 2: Solve the right-hand side \( \cot(\cos^{-1}(x)) \)

1. **Construct another right triangle** where the angle \( \phi = \cos^{-1}(x) \).
- Here, the adjacent side is \( x \) and the hypotenuse is \( 1 \).
2. **Find the opposite side** using the Pythagorean theorem:
\[ \text{opposite} = \sqrt{1 - x^2} \]
3. **Calculate \( \cot \phi \)**:
\[ \cot(\phi) = \frac{\text{adjacent}}{\text{opposite}} = \frac{x}{\sqrt{1 - x^2}} \]

### Step 3: Set the two sides equal

Now we equate the two expressions we found:
\[ \frac{1}{\sqrt{5}} = \frac{x}{\sqrt{1 - x^2}} \]

### Step 4: Cross-multiply and simplify

Cross-multiplying gives:
\[ 1 \cdot \sqrt{1 - x^2} = x \cdot \sqrt{5} \]
Squaring both sides results in:
\[ 1 - x^2 = 5x^2 \]

### Step 5: Rearrange the equation

Rearranging gives:
\[ 1 = 6x^2 \]

### Step 6: Solve for \( x \)

Dividing both sides by 6:
\[ x^2 = \frac{1}{6} \]
Taking the square root gives:
\[ x = \pm \frac{1}{\sqrt{6}} \]

### Step 7: Determine the valid value of \( x \)

Since \( x \) must be in the range of the cosine function (which is between -1 and 1), we check both values:
- \( x = \frac{1}{\sqrt{6}} \) is valid.
- \( x = -\frac{1}{\sqrt{6}} \) is not valid because it would make the cotangent negative, which does not match the positive value we found on the left side.

Thus, the final value of \( x \) is:
\[ \boxed{\frac{1}{\sqrt{6}}} \]