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:
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