Find exact solutions of the equation cos(𝑥)=1−cos(𝑥) for 𝑥 in the interval [0,2𝜋].

Sure, let's solve the equation step by step.

1. First, we have the equation cos(x) = 1 - cos(x).

2. To simplify, we can move all terms to one side of the equation. This gives us 2cos(x) - 1 = 0.

3. Now, we can solve for cos(x) by adding 1 to both sides and then dividing by 2. This gives us cos(x) = 1/2.

4. Now, we need to find the values of x in the interval [0, 2π] that satisfy this equation.

5. The cosine function has a value of 1/2 at two points within this interval: π/3 and 5π/3.

6. Therefore, the solutions to the equation cos(x) = 1 - cos(x) in the interval [0, 2π] are x = π/3 and x = 5π/3.