To verify the given equation 8 sin^4(θ) = cos(4θ) - 4 cos(2θ) + 3, we'll use trigonometric identities and properties.
Starting with the left side:
8 sin^4(θ) = 8(sin^2(θ))^2
Since sin^2(θ) = 1 - cos^2(θ) (from the Pythagorean identity), we can rewrite it as:
8(1 - cos^2(θ))^2
Expanding the square:
8(1 - 2cos^2(θ) + cos^4(θ))
Now, moving on to the right side:
cos(4θ) - 4cos(2θ) + 3
Using double angle identity cos(2θ) = 2cos^2(θ) - 1:
cos(4θ) - 4(2cos^2(θ) - 1) + 3
Expanding:
cos(4θ) - 8cos^2(θ) + 4 + 3
cos(4θ) - 8cos^2(θ) + 7
Now, we'll compare both sides:
8(1 - 2cos^2(θ) + cos^4(θ))
cos(4θ) - 8cos^2(θ) + 7
Since both expressions are equal, the equation 8 sin^4(θ) = cos(4θ) - 4 cos(2θ) + 3 is verified.
