Given:
We are given the following equations:
1) xy = 1/4
2) cot(x) * cot(y) = 2

To find:
We need to find the smallest positive values of x and y satisfying the given equations.

Solution:
To solve the given equations, we can use the following steps:

Step 1: Simplify the second equation using the identity cot(x) * cot(y) = 1/tan(x) * 1/tan(y) = 1/(tan(x) * tan(y)).

Step 2: Substitute the value of xy from the first equation into the second equation to get 1/(tan(x) * tan(y)) = 2.

Step 3: Cross-multiply to get tan(x) * tan(y) = 1/2.

Step 4: Use the identity tan(x) * tan(y) = (tan(x) + tan(y))/(1 - tan(x) * tan(y)).

Step 5: Substitute the value of tan(x) * tan(y) = 1/2 into the above identity to get (tan(x) + tan(y))/(1 - 1/2) = (tan(x) + tan(y))/(1/2) = 2(tan(x) + tan(y)) = 1/2.

Step 6: Simplify the equation to get tan(x) + tan(y) = 1/4.

Step 7: Use the identity tan(x + y) = (tan(x) + tan(y))/(1 - tan(x) * tan(y)).

Step 8: Substitute the values of tan(x) + tan(y) = 1/4 and tan(x) * tan(y) = 1/2 into the above identity to get tan(x + y) = (1/4)/(1 - 1/2) = 1/4.

Step 9: Since tan(x + y) = tan(π/2), we can write x + y = π/2.

Step 10: To find the smallest positive values of x and y, we need to consider the smallest positive values of x + y. Since x + y = π/2, the smallest positive values of x and y are x = π/6 and y = 5π/12.

Answer:
The smallest positive values of x and y satisfying the given equations are x = π/6 and y = 5π/12. However, the answer options are given in terms of fractions, so we need to convert the values of x and y into fractions. Using the fact that π/6 = 30° and 5π/12 = 75°, we can convert the values into fractions as x = 5/12 and y = π/6. Therefore, the correct answer is option A) x = 5/12 and y = π/6.