How many values of θ with 0° ≤ θ ≤ 360° satisfy cosθ = 1?

The cosine reaches 1 exactly at whole multiples of 360°. The interval here is closed at both ends, so θ = 0° and θ = 360° both count, giving 2 values. Had the range been written 0° ≤ θ < 360°, only one value would qualify — the endpoint is the whole point of the question.