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We can solve this equation by utilizing the Pythagorean identity, which states that cos^2(x) + sin^2(x) = 1.

Here's how we proceed:

Rewrite the equation: We can rewrite the given equation as cos^2(x) - sin^2(x) - sin(x) = 0.

Factor the equation: Noticing a common factor of cos^2(x) - 1, we can factor the equation as (cos^2(x) - 1) * (sin(x) + 1) = 0.

Separate the factors: This gives us two possibilities:

cos^2(x) - 1 = 0

sin(x) + 1 = 0

Solve for cos(x):

From cos^2(x) - 1 = 0, add 1 to both sides and take the square root of both sides: cos(x) = ±1. Since 0 < x <= 2*pi, the solutions for cos(x) are x = 0 and x = pi.

Solve for sin(x):

From sin(x) + 1 = 0, subtract 1 from both sides: sin(x) = -1. However, within the range 0 < x <= 2*pi, there is no solution for sin(x) to be -1.

Combine solutions: Therefore, the solutions for the equation within the given range are x = 0 and x = pi.

Answer: The solutions for the equation cos^2(x) - sin^2(x) = sin(x) within the range 0 < x <= 2*pi are x = 0 and x = pi.

Boseo Jul 8, 2024