To solve this equation, we can use the trigonometric identities:
Using these identities, we can rewrite the given equation as:
(sin(x/2)^2)^2 - (cos(x/2)^2)^2 = (sin^2(x/2) + cos^2(x/2))(sin^2(x/2) - cos^2(x/2)) = (1)(sin^2(x/2) - cos^2(x/2)) = 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 - 2cos^2(x/2) = 1
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.
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))^2Using these identities, we can rewrite the given equation as:
(sin(x/2)^2)^2 - (cos(x/2)^2)^2 =
(sin^2(x/2) + cos^2(x/2))(sin^2(x/2) - cos^2(x/2)) =
(1)(sin^2(x/2) - cos^2(x/2)) =
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) =
2cos^2(x/2) =1 - 2cos^2(x/2) = 1
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.