STRATEGY: Upon reading any GRE multiple choice question, we should always ask, Can I use the answer choices to my advantage?
In this case, we can easily test values of x and y.
Now let's give ourselves up to 20 seconds to identify a faster approach.
In this case, we might try some algebraic manipulation, but it's not 100% apparent that this strategy will work. So I'll stick with the strategy I know will work...

APPROACH #1: Testing values that satisfy the given information
Since x + y = a and x - y = b, it could be the case that x = 3 and y = 2, in which case 2xy = 2(3)(2) = 12
So, we're looking for an answer choice that evaluates to 12

Since the answer choices are terms of a and b, we need to find the corresponding values of a and b when x = 3 and y = 2
So, we'll plug x = 3 and y = 2 into the equations x + y = a and x - y = b

Take: x + y = a
Substitute values to get: 3 + 2 = 5, which means a = 5

Take: x - y = a
Substitute values to get: 3 - 2 = 1, which means b = 1

Now plug a = 5 and b = 1 into the 5 answer choices to see which one(s) evaluate to 12...

A. \(\frac{(5^2 - 1^2)}{2} = \) 12. KEEP!

B. \(\frac{(1^2 - 5^2)}{2}=\) -12. Eliminate

C. \(\frac{(5 - 1 )}{2}=\) 2. Eliminate

D. \(\frac{(5)(1)}{2}=\) 2.5. Eliminate

E. \(\frac{(5^2 + 1^2)}{2}=\) 13. Eliminate

By the process of elimination, the correct answer is A.

APPROACH #2: Algebraic manipulation
Given:
x + y = a
x - y = b
Add the two equations to get: 2x = a + b

Similarly, we can take the two equations....
x + y = a
x - y = b
.... And subtract the bottom equation from the top equation to get: 2y = a - b

This means we can write: (2x)(2y) = (a + b)(a - b)
Expand and simplify both sides: 4xy = a² - b²
Divide both sides of the equation by 2 to get: 2xy = (a² - b²)/2

Answer: A