In Exercises 63–84, use an identity to solve each equation on the interval [0, 2𝝅). cos 2x = cos x

Verified step by step guidance

1. Recognize that the equation $\cos 2x=\cos x$ can be solved using trigonometric identities.
2. Use the double angle identity for cosine: $\cos 2x=2{\cos }^{2}x-1$.
3. Substitute the identity into the equation: $2{\cos }^{2}x-1=\cos x$.
4. Rearrange the equation to form a quadratic equation: $2{\cos }^{2}x-\cos x-1=0$.
5. Solve the quadratic equation for $\cos x$ using the quadratic formula or factoring, and find the solutions for $x$ in the interval $[0,2\pi )$.

Key Concepts

Trigonometric Identities

Trigonometric identities are equations that involve trigonometric functions and are true for all values of the variables involved. In this case, the double angle identity for cosine, cos(2x) = 2cos²(x) - 1, can be used to rewrite the equation. Understanding these identities is crucial for simplifying and solving trigonometric equations.

Solving Trigonometric Equations

Solving trigonometric equations involves finding the values of the variable that satisfy the equation. This often requires using identities to transform the equation into a more manageable form. In this problem, we need to manipulate the equation to isolate x and find all solutions within the specified interval [0, 2π).

Interval Notation

Interval notation is a way of describing a range of values, often used in mathematics to specify the domain or range of functions. The interval [0, 2π) indicates that we are looking for solutions starting from 0 up to, but not including, 2π. Understanding this notation is essential for determining the valid solutions to the equation within the specified bounds.