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Baltic Way competition problems

Algebra Difficulty 4.3 AIME Prove it Baltic Way

Let $x$ be a positive acute angle. Prove that
$\cos^2(x) \cot(x) + \sin^2(x) \tan(x) \ge 1$

Solution

The geometric-arithmetic inequality gives
$\cos x \sin x \le \frac{\cos^2 x + \sin^2 x}{2} = \frac{1}{2}.$
It follows that
$1 = (\cos^2 x + \sin^2 x)^2 = \cos^4 x + \sin^4 x + 2 \cos^2 x \sin^2 x \le \cos^4 x + \sin^4 x + \frac{1}{2}$
so
$\cos^4 x + \sin^4 x \ge \frac{1}{2} \ge \cos x \sin x.$
The required inequality follows.

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Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.