Explanation:

To determine whether the function f(x) = tan⁻¹(sin(x)cos(x)) is increasing or decreasing, we need to analyze the sign of its derivative.

Derivative of f(x):
Let's find the derivative of f(x) using the chain rule and product rule.

f'(x) = d(tan⁻¹(sin(x)cos(x)))/dx
= (1/(1 + (sin(x)cos(x))²)) * d(sin(x)cos(x))/dx

Using the product rule, we can differentiate sin(x)cos(x) as follows:

d(sin(x)cos(x))/dx = cos²(x) - sin²(x)

Substituting this back into the derivative of f(x), we get:

f'(x) = (1/(1 + (sin(x)cos(x))²)) * (cos²(x) - sin²(x))
= (cos²(x) - sin²(x))/(1 + (sin(x)cos(x))²)

Analyzing the sign of f'(x):
To determine whether f(x) is increasing or decreasing, we need to analyze the sign of f'(x) in the given interval.

Let's consider the interval (-π/2, π/4).

1. Analyzing the numerator: (cos²(x) - sin²(x))
In the given interval, both cos²(x) and sin²(x) are positive because cos(x) and sin(x) are positive. Therefore, the numerator is positive.

2. Analyzing the denominator: (1 + (sin(x)cos(x))²)
In the given interval, sin(x)cos(x) is positive because both sin(x) and cos(x) are positive. Therefore, (sin(x)cos(x))² is positive. Adding 1 to a positive number gives a positive denominator.

Conclusion:
Since the numerator of f'(x) is positive and the denominator is positive, the function f(x) = tan⁻¹(sin(x)cos(x)) is always positive in the interval (-π/2, π/4). This means that the function is increasing in this interval.

Therefore, the correct answer is option 'B': (-π/2, π/4).