Let's first simplify the expression using trigonometric identities:

cos^4(5π/12) - sin^4(5π/12)

Using the Pythagorean identity sin^2x + cos^2x = 1, we can rewrite the expression as:

(cos^2(5π/12))^2 - (1 - cos^2(5π/12))^2

Expanding the squares:

(cos^2(5π/12))^2 - (1 - 2cos^2(5π/12) + cos^4(5π/12))

Now, simplify further:

cos^4(5π/12) - 1 + 2cos^2(5π/12) - cos^4(5π/12)

The cos^4(5π/12) and -cos^4(5π/12) terms cancel out, leaving:

2cos^2(5π/12) - 1

Therefore, the simplified expression is:

2cos^2(5π/12) - 1