To simplify the expression cos^2(x-pi/5)-sin^2(2x+4pi/5), we can use the trigonometric identity:

cos^2(x) - sin^2(x) = cos(2x)

Applying this identity, we have:

cos^2(x-pi/5)-sin^2(2x+4pi/5) = cos(2(x-pi/5)) - sin^2(2x+4pi/5)

Now, let's simplify each term separately:

cos(2(x-pi/5)) = cos(2x - 2pi/5)

Using the double angle formula for cosine, we have:

cos(2x - 2pi/5) = cos^2(x - pi/5) - sin^2(x - pi/5)

Next, let's simplify the second term:

sin^2(2x+4pi/5) = sin^2(2x + 4pi/5)

Using the double angle formula for sine, we have:

sin^2(2x + 4pi/5) = 1 - cos^2(2x + 4pi/5)

Now, let's substitute these simplified terms back into the original expression:

cos^2(x-pi/5)-sin^2(2x+4pi/5) = cos^2(x - pi/5) - sin^2(x - pi/5) - (1 - cos^2(2x + 4pi/5))

Expanding the expression further:

= cos^2(x - pi/5) - sin^2(x - pi/5) - 1 + cos^2(2x + 4pi/5)

Now, we can combine like terms:

= cos^2(x - pi/5) + cos^2(2x + 4pi/5) - sin^2(x - pi/5) - 1

Finally, we can use the Pythagorean identity sin^2(x) + cos^2(x) = 1 to simplify further:

= cos^2(x - pi/5) + cos^2(2x + 4pi/5) - (1 - cos^2(x - pi/5))

= cos^2(x - pi/5) + cos^2(2x + 4pi/5) - cos^2(x - pi/5) + 1

= cos^2(2x + 4pi/5) + 1

Therefore, the simplified expression is cos^2(2x + 4pi/5) + 1.