To solve this equation, we can simplify it step by step.

First, let's simplify the terms involving cosine and sine:

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

Next, let's simplify the terms involving cosine and sine:

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

Now, let's focus on the term involving cosine:

-sin x * cos 4x - cos^2 (4x)

Using the identity cos^2 (4x) = 1 - sin^2 (4x), we can rewrite the term as:

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

Expanding the term, we get:

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

Now, let's substitute this back into the original equation:

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

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

Simplifying further, we get:

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

Now, we have a quadratic equation in terms of sin x:

sin^2 (4x) - sin x * cos 4x + 1/4 = 0

To solve this quadratic equation, we can use the quadratic formula:

sin x = [-(-cos 4x) ± √((-cos 4x)^2 - 4(1/4)(1))]/(2(1/4))

sin x = [cos 4x ± √(cos^2 4x - 1)]/1/2

sin x = 2(cos 4x ± √(cos^2 4x - 1))

Now, we can solve for sin x and substitute it back into the original equation to find the values of x that satisfy the equation.