Final answer:
The equation cos(2x) = cos(x) - 1 can be solved using the double angle identity for cosine. By comparing the given equation with the double angle formulas, specifically 2cos^2(x) - 1, a quadratic equation is obtained in terms of cos(x) and can be solved to find x.
Step-by-step explanation:
The student asked to use an identity to solve the equation cos(2x) = cos(x) - 1. By utilizing trigonometric identities, we can solve this equation. The double angle formula for cosine can be written as cos(2x) = 2cos2(x) - 1 or as cos(2x) = 1 - 2sin2(x). By comparing these with the given equation, we can see that 2cos2(x) - 1 fits our needs since the equation resembles the first form of the double angle identity.
Therefore, we set up the equation:
- cos(2x) = 2cos2(x) - 1,
- 2cos2(x) - cos(x) - 1 = 0.
This is a quadratic equation in terms of cos(x) which can be factored or solved using the quadratic formula to find the solutions for x.