To solve this equation, we can use the trigonometric identities:

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

Using these identities, we can rewrite the given equation as:

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

Now, using the identity sin^2(x) = 1 - cos^2(x), we can rewrite the equation as:

(1 - cos^2(x/2)) - cos^2(x/2) = 1
1 - 2cos^2(x/2) = 1

2cos^2(x/2) = 0
cos^2(x/2) = 0

Taking the square root of both sides, we get:

cos(x/2) = 0

This means that x/2 must be equal to pi/2 or 3pi/2:

x/2 = pi/2 or x/2 = 3pi/2
x = pi or x = 3pi

Therefore, the solutions to the equation sin(x/2)^4 - cos(x/2)^4 = 1 are x = pi or x = 3pi.