The key concept here is that \(\frac{1}{(-1)^x} = (-1)^x\). Here's why:
Since x must be either odd or even, we can consider both cases.

If x is ODD, then \(\frac{1}{(-1)^x} = \frac{1}{-1} = -1\)
If x is ODD, then \((-1)^x = -1\)
So, when x is ODD, \(\frac{1}{(-1)^x} = (-1)^x\)

If x is EVEN, then \(\frac{1}{(-1)^x} = \frac{1}{1} = 1\)
If x is EVEN, then \((-1)^x = 1\)
So, when x is EVEN, \(\frac{1}{(-1)^x} = (-1)^x\)

Since we have analyzed both possible cases, we can be certain that \(\frac{1}{(-1)^x} = (-1)^x\), which means...
QUANTITY A: \(\frac{1}{(-1)^x}-(-1)^x = 0\)
QUANTITY B: \(0\)

Answer: C