Solution: Expand the expression: $ \cos^2x + 2 + \sec^2x + \sin^2x + 2 + \csc^2x $. Simplify using $ \cos^2x + \sin^2x = 1 $ and $ \sec^2x = 1 + an^2x $, $ \csc^2x = 1 + \cot^2x $: $ 1 + 2 + 1 + an^2x + 1 + \cot^2x + 2 = 7 + an^2x + \cot^2x $. Let $ t = an^2x $, so $ \cot^2x = rac{1}{t} $. The expression becomes $ 7 + t + rac{1}{t} $. By AM-GM, $ t + rac{1}{t} \geq 2 $, so the minimum is $ 7 + 2 = 9 $. Thus, the minimum value is $ oxed{9} $.

Tue, 05 May 2026 01:57:20 +0000

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