What is the difference between the <strong>GREATEST</strong> value and the least value of \cos^{2}\theta+3 \sin^{2}\theta+2?

Explanation

Rewrite the expression as (1 - \sin^2 \theta) + 3 \sin^2 \theta + 2 = 3 + 2 \sin^2 \theta. The minimum value is 3 + 0 = 3 (when \sin \theta = 0), and the maximum value is 3 + 2(1) = 5 (when \sin^2 \theta = 1). The difference is 5 - 3 = 2.

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