A Four-Fraction Symmetry

★ 8/10trigonometryCauchy-Schwarzinequality

Problem

For x∈(0,π2)x \in \left(0, \dfrac{\pi}{2}\right), find the minimum value of

sin⁡x+cos⁡xsin⁡x+tan⁡x+tan⁡x+cot⁡xcos⁡x+tan⁡x+sin⁡x+cos⁡xcos⁡x+cot⁡x+tan⁡x+cot⁡xsin⁡x+cot⁡x.\frac{\sin x + \cos x}{\sin x + \tan x} + \frac{\tan x + \cot x}{\cos x + \tan x} + \frac{\sin x + \cos x}{\cos x + \cot x} + \frac{\tan x + \cot x}{\sin x + \cot x}.

Answer

Solution

Difficulty8/10
Topicstrigonometry, Cauchy-Schwarz, inequality

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