Solve the equation on the interval: 0 ≤ x ≤ 2π for 2 cos(x) – 2 cos(x) sin(x) = 0

To solve the equation 2 cos(x) – 2 cos(x) sin(x) = 0, we can start by factoring:

Factor out 2 cos(x):

2 cos(x) (1 – sin(x)) = 0

This gives us two cases to consider:

1. 2 cos(x) = 0
2. 1 – sin(x) = 0

For the first case, 2 cos(x) = 0:

Dividing by 2, we have cos(x) = 0. The solutions for this equation in the interval [0, 2π] are:

• x = π/2
• x = 3π/2

For the second case, 1 – sin(x) = 0:

This simplifies to sin(x) = 1. The solution for this equation in the interval [0, 2π] is:

• x = π/2

Now, we combine the solutions from both cases. The results are:

• x = π/2 (from both cases)
• x = 3π/2 (from the first case only)

Thus, the complete set of solutions to the equation 2 cos(x) – 2 cos(x) sin(x) = 0 on the interval [0, 2π] is:

• x = π/2
• x = 3π/2