If `tan(cos^(-1) x) = sin (cot^(-1).(1)/(2))`, then find the value of x

Explanation

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

### Step 1: Rewrite the left side
Let \( \theta = \cos^{-1} x \). Then, we have:
\[ x = \cos \theta \]
Thus, the left side becomes:
\[ \tan(\cos^{-1} x) = \tan(\theta) \]

### Step 2: Express \( \tan(\theta) \)
Using the right triangle definition, where the adjacent side is \( x \) (base) and the hypotenuse is \( 1 \), we can find the opposite side (perpendicular):
\[ \text{Perpendicular} = \sqrt{1^2 - x^2} = \sqrt{1 - x^2} \]
Thus,
\[ \tan(\theta) = \frac{\text{Perpendicular}}{\text{Base}} = \frac{\sqrt{1 - x^2}}{x} \]

### Step 3: Rewrite the right side
Now, consider the right side \( \sin(\cot^{-1} \frac{1}{2}) \). Let \( \alpha = \cot^{-1} \frac{1}{2} \). Then,
\[ \cot \alpha = \frac{1}{2} \]
This means the adjacent side (base) is \( 1 \) and the opposite side (perpendicular) is \( 2 \).

### Step 4: Find the hypotenuse for \( \alpha \)
Using the Pythagorean theorem:
\[ \text{Hypotenuse} = \sqrt{(1)^2 + (2)^2} = \sqrt{1 + 4} = \sqrt{5} \]
Thus, we can express \( \sin(\alpha) \):
\[ \sin(\alpha) = \frac{\text{Perpendicular}}{\text{Hypotenuse}} = \frac{2}{\sqrt{5}} \]

### Step 5: Set the two sides equal
Now, we equate the two sides:
\[ \frac{\sqrt{1 - x^2}}{x} = \frac{2}{\sqrt{5}} \]

### Step 6: Cross-multiply
Cross-multiplying gives:
\[ \sqrt{5} \sqrt{1 - x^2} = 2x \]

### Step 7: Square both sides
Squaring both sides results in:
\[ 5(1 - x^2) = 4x^2 \]

### Step 8: Rearrange the equation
Rearranging gives:
\[ 5 - 5x^2 = 4x^2 \]
\[ 5 = 9x^2 \]

### Step 9: Solve for \( x^2 \)
Thus, we have:
\[ x^2 = \frac{5}{9} \]

### Step 10: Find \( x \)
Taking the square root gives:
\[ x = \pm \frac{\sqrt{5}}{3} \]

### Step 11: Determine the valid solution
Since \( \cos^{-1} x \) is defined for \( x \) in the range \([-1, 1]\), we discard \( x = -\frac{\sqrt{5}}{3} \) because it would yield a negative value for \( \tan(\theta) \). Therefore, the valid solution is:
\[ x = \frac{\sqrt{5}}{3} \]

### Final Answer
The value of \( x \) is:
\[ \boxed{\frac{\sqrt{5}}{3}} \]