Given that cos(θ) = 0 and 0 < θ < π/2, we need to find the value of (sin(θ) - cos(θ)) / (2 * sin(θ) * cos(θ) * tan(θ)).

First, since cos(θ) = 0 and 0 < θ < π/2, there seems to be a contradiction, because cos(θ) is not 0 within the interval (0, π/2). Cosine is zero at π/2 (90 degrees). Assuming that θ approaches π/2, let's examine the limit.

If cos(θ) = 0, then θ = π/2. Then sin(θ) = sin(π/2) = 1. Also, tan(θ) = sin(θ) / cos(θ). But tan(π/2) is undefined because division by zero is undefined. So we can't simply substitute the values directly.

Let's analyze the expression: (sin(θ) - cos(θ)) / (2 * sin(θ) * cos(θ) * tan(θ)) = (sin(θ) - cos(θ)) / (2 * sin(θ) * cos(θ) * (sin(θ) / cos(θ))) = (sin(θ) - cos(θ)) / (2 * sin²(θ))

Now, as θ approaches π/2: sin(θ) approaches 1 cos(θ) approaches 0 So, the expression becomes: (1 - 0) / (2 * 1²) = 1 / 2

However, since the initial problem has cos(θ) = 0 while 0 < θ < π/2, this is not strictly correct.

Let's assume cos(θ) is approximately zero, and the value of theta can be π/2. If that's the intended meaning:

(sin(θ) - cos(θ)) / (2 * sin(θ) * cos(θ) * tan(θ)) = (1 - 0) / (2 * 1 * 0 * tan(π/2)) This is undefined.

However, simplifying the equation algebraically gives us the expression: (sin(θ) - cos(θ)) / (2 * sin²(θ)). As θ approaches π/2, this is approximately equal to 1/2.

Given the initial condition's potential inconsistency, and relying on the simplification and limit approach, the value is 1/2.