Here is the correct solution:
Given that sin x = 5/13, we can use the Pythagorean identity cos²x + sin²x = 1 to find cos x as follows:
cos²x + (5/13)² = 1
cos²x = 1 - (5/13)²
cos x = ±sqrt(1 - (5/13)²)
Since 0° < x < 90°, we know that cos x > 0. Therefore, we can take the positive square root:
cos x = sqrt(1 - (5/13)²)
cos x = 12/13
Next, we can evaluate sin x / (2tan x) as follows:
sin x / (2tan x) = (5/13) / (2sin x / cos x)
sin x / (2tan x) = (5/13) * (cos x / 2sin x)
sin x / (2tan x) = (5/13) * (cos x / (2 * (5/13)))
sin x / (2tan x) = cos x / 2
Substituting the value of cos x that we found earlier, we get:
sin x / (2tan x) = (12/13) / 2
sin x / (2tan x) = 6/13
Finally, we can evaluate cos x - sin x / (2tan x) as follows:
cos x - sin x / (2tan x) = (12/13) - (5/13) / (2 * (5/13))
cos x - sin x / (2tan x) = 12/13 - 1/13
cos x - sin x / (2tan x) = 11/13
Therefore, cos x - sin x / (2tan x) = 11/13.