Step-by-Step Solution

1. Step 1: Differentiate the function \(f(x)\) with respect to \(x\).

   Given \(f(x) = a(\tan x - \cot x)\), we need to find \(f'(x)\).

   \(f'(x) = a(\sec^2 x - (-\csc^2 x)) = a(\sec^2 x + \csc^2 x)\)

2. Step 2: Analyze the sign of \(f'(x)\).

   Since \(a > 0\), \(\sec^2 x\) is always positive, and \(\csc^2 x\) is always positive, their sum is also always positive.

   Therefore, \(f'(x) = a(\sec^2 x + \csc^2 x) > 0\) for all \(x\) in the domain of \(f(x)\).

3. Step 3: Conclude whether \(f(x)\) is increasing or decreasing.

   Since \(f'(x) > 0\) for all \(x\) in the domain, the function \(f(x)\) is an increasing function.