The difference between the greatest and the least possible value of the expression 3 - cos ⁡ x + sin 2 ⁡ x is

Step-by-step solution

3 - cos ⁡ x + 1 - cos 2 ⁡ x = - cos 2 ⁡ x + cos ⁡ x + 4 = - cos 2 ⁡ x + cos ⁡ x + 1 4 + 4 + 1 4 = - cos ⁡ x + 1 2 2 + 17 4 = 17 4 - cos ⁡ x + 1 2 2 Maximum value at   cos ⁡ x = - 1 2 = 17 4 Minimum value at   cos ⁡ x = 1 = 17 4 - 9 4 = 2 Difference = 17 4 - 2 = 9 4