The difference between the greatest and the least possible value of the expression $3 - \cos a + \sin^2 a$ is
Step-by-Step Solution
Key Concept: Rewrite a trigonometric expression in the form of a perfect square to find its maximum and minimum values using the range of $\cos x \in [-1, 1]$.
Rewrite the expression as $3 - \cos x + 1 - \cos^2 x = (\cos^2 x + \cos x) + 4 = -(\cos^2 x + \cos x - \frac{1}{2}) + 4 + \frac{1}{2} = -(\cos x + \frac{1}{2})^2 + \frac{17}{4}$. The maximum occurs when $\cos x = -\frac{1}{2}$, giving maximum value $\frac{17}{4}$. The minimum occurs when $\cos x = 1$, giving minimum value $1 - 1 - 1 = -1$ or at $\cos x = -1$ giving $-1 - 1 + 1 = -1 - \frac{1}{2} = -\frac{1}{2}$. The difference between maximum $\frac{17}{4}$ and minimum $-2$ is $\frac{17}{4} - (-2) = \frac{17}{4} + 2 = \frac{25}{4}$.
Correct Answer: 2