Solution: Since $ \sin^2 x + \cos^2 x = 1 $, we write $ f(x) = \frac{1}{\sin^2 x + 4\cos^2 x} = \frac{1}{1 + 3\cos^2 x} $. Let $ y = \cos^2 x $, so $ 0 \leq y \leq 1 $. Then $ f(x) = \frac{1}{1 + 3y} $. As $ y $ varies from 0 to 1, $ 1 + 3y $ varies from 1 to 4, so $ f(x) $ varies from $ \frac{1}{4} $ to $ 1 $. Hence, the range of $ f(x) $ is $ \left[\frac{1}{4}, 1\right] $. The final answer is $ \boxed{\left[\frac{1}{4}, 1\right]} $.

Wed, 06 May 2026 18:44:40 +0000

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