To solve this equation, we can start by simplifying the expression on the left side:

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

Using the identity cos^2 x = 1 - sin^2 x, we can rewrite the equation as:

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

Expanding the square of cosine (cos^2 (4x) = 1 - sin^2 (4x)), we get:

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

Simplifying further, we have:

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

Combining like terms, we get:

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

Using the identity sin^2 x = 1 - cos^2 x, we can rewrite the equation as:

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

Expanding the square of sine (sin^2 (4x) = 1 - cos^2 (4x)), we get:

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

Simplifying further, we have:

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

Combining like terms, we get:

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

This is the same equation we started with. Therefore, the equation is true for all values of x.