To solve the equation cos(x - pi/5)^2 = sin(2*x + 4*pi/5)^2, we can start by using the trigonometric identity:

sin^2(a) + cos^2(a) = 1

Applying this identity to both sides of the equation, we get:

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

Now, let's simplify the equation further. Using the double angle formula for sine, sin(2x) = 2sin(x)cos(x), we can rewrite the equation as:

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

Expanding the square of the second term, we have:

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

Using the identity sin^2(a) = 1 - cos^2(a), we can substitute sin^2(x + 2*pi/5) = 1 - cos^2(x + 2*pi/5) into the equation:

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

Simplifying further, we get:

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

Now, let's combine like terms:

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

Rearranging the terms, we have:

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

Now, we can simplify the equation further by factoring out cos^2(x + 2*pi/5):

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

Since cos^2(x + 2*pi/5) cannot be zero (otherwise, the equation would not hold), we can divide both sides of the equation by cos^2(x + 2*pi/5):

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

Simplifying further, we get:

-4cos^2(x + 2*pi/5) + cos^2(x - pi/5) = -3

Now, we can solve this equation for x. However, it is important to note that this equation does not have a simple algebraic solution. It may require numerical methods or approximation techniques to find the values of x that satisfy the equation.