Step 1: Recognise the mixture.
Acetic acid is a weak acid and sodium acetate is its conjugate salt, so mixing them produces an acidic buffer solution.
Step 2: Recall the Henderson equation.
For an acidic buffer, $pH = pK_a + \log\dfrac{[\text{salt}]}{[\text{acid}]}$.
Step 3: Compare the amounts mixed.
Equal volumes of $0.5\,N$ acid and $0.5\,N$ salt are mixed, so after mixing both are diluted by the same factor and remain equal in concentration.
Step 4: Form the ratio.
Hence $\dfrac{[\text{salt}]}{[\text{acid}]} = 1$.
Step 5: Evaluate the logarithm.
Since $\log 1 = 0$, the equation becomes $pH = pK_a + 0 = 4.75$.
Step 6: State the answer.
The pH of the resultant solution is $4.75$, which is option (3). \[ \boxed{4.75} \]