Final answer:
To solve the equation cos(2θ) = 2cos(θ) - 2cos²(θ), we can use the double-angle identity for cosine. We simplify the equation by combining like terms and then apply the Pythagorean identity for sine and cosine. Finally, we rearrange the equation as a quadratic equation to find the values of θ.
Step-by-step explanation:
To solve the equation cos(2θ) = 2cos(θ) - 2cos²(θ), we can use the double-angle identity for cosine. The double-angle identity states that cos(2θ) = cos²(θ) - sin²(θ). So, we can rewrite the equation as: cos²(θ) - sin²(θ) = 2cos(θ) - 2cos²(θ).
Next, we can simplify the equation by combining like terms. cos²(θ) - 2cos²(θ) + sin²(θ) = 2cos(θ). This simplifies to -cos²(θ) + sin²(θ) = 2cos(θ).
Finally, we can apply the Pythagorean identity for sine and cosine, which states that sin²(θ) + cos²(θ) = 1. By substituting this identity into our equation, we get -cos²(θ) + (1 - cos²(θ)) = 2cos(θ). Simplifying further, we get -2cos²(θ) + 1 = 2cos(θ).
To solve for θ, we can rearrange the equation as a quadratic equation: -2cos²(θ) - 2cos(θ) + 1 = 0. We can then solve this quadratic equation to find the values of θ that satisfy the equation.