Question

Use a double-angle identity to solve cos²x - sin² x =1/2 in [0,2π)

A. x=π/6
B. x=π/4
C. x=π/8
D.x=π/2

Answer 1

To solve the equation cos²x - sin²x = 1/2 in the interval [0,2π), use the double-angle identity for cosine. Rewriting the equation and eliminating solutions that do not fall within the interval, x = π/3 is a solution.

Step-by-step explanation:

To solve the equation cos²x - sin²x = 1/2 in the interval [0,2π), we can use the double-angle identity for sine and cosine. The double-angle identity states that sin(2θ) = 2sin(θ)cos(θ) and cos(2θ) = cos²(θ) - sin²(θ). Using the double-angle identity for cosine, we can rewrite the equation as 2cos²(x) - 1 = 1/2. Simplifying this expression, we get 2cos²(x) = 3/2. Dividing both sides by 2, we have cos²(x) = 3/4. Taking the square root of both sides, we get cos(x) = ±√(3/4). Since we are only interested in solutions in the interval [0, 2π), we need to find x values that satisfy cos(x) = √(3/4) and eliminate any values that do not fall within that interval.

Since cos(x) represents the x-coordinate on the unit circle, the positive square root (√(3/4)) corresponds to the cosine value of π/3. Therefore, x = π/3 is a solution in the interval [0, 2π).