Evaluate the following expression:
`cos tan^(-1)sin cot^(-1)(1/2)`

Explanation

To evaluate the expression \( \cos(\tan^{-1}(\sin(\cot^{-1}(1/2)))) \), we will follow these steps:

### Step 1: Evaluate \( \cot^{-1}(1/2) \)
Let \( \theta = \cot^{-1}(1/2) \). This implies that \( \cot(\theta) = \frac{1}{2} \).

Using the definition of cotangent:
\[ \cot(\theta) = \frac{\text{adjacent}}{\text{opposite}} = \frac{1}{2} \]
We can represent this in a right triangle where the adjacent side is 1 and the opposite side is 2.

### Step 2: Find the hypotenuse
Using the Pythagorean theorem:
\[ \text{hypotenuse} = \sqrt{(\text{adjacent})^2 + (\text{opposite})^2} = \sqrt{1^2 + 2^2} = \sqrt{1 + 4} = \sqrt{5} \]

### Step 3: Find \( \sin(\theta) \)
Now, we can find \( \sin(\theta) \):
\[ \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{2}{\sqrt{5}} \]

### Step 4: Substitute into the expression
Now we substitute \( \sin(\theta) \) back into the expression:
\[ \cos(\tan^{-1}(\sin(\cot^{-1}(1/2)))) = \cos(\tan^{-1}(\frac{2}{\sqrt{5}})) \]

### Step 5: Let \( y = \tan^{-1}(\frac{2}{\sqrt{5}}) \)
This implies:
\[ \tan(y) = \frac{2}{\sqrt{5}} \]

### Step 6: Find the hypotenuse for \( y \)
Using the same triangle approach:
- Opposite = 2
- Adjacent = \( \sqrt{5} \)

Using the Pythagorean theorem again:
\[ \text{hypotenuse} = \sqrt{(2)^2 + (\sqrt{5})^2} = \sqrt{4 + 5} = \sqrt{9} = 3 \]

### Step 7: Find \( \cos(y) \)
Now we find \( \cos(y) \):
\[ \cos(y) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{\sqrt{5}}{3} \]

### Final Result
Thus, the final value of the expression is:
\[ \cos(\tan^{-1}(\sin(\cot^{-1}(1/2)))) = \frac{\sqrt{5}}{3} \]