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ring of roses nursery rhyme - Red Crowns

📅 March 11, 2026 👤 scraface

Mar 11, 2026

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📌 We expand both terms:

📌 \sin^2 x + \cos^2 x + \sec^2 x + \csc^2 x + 2(\tan x + \cot x)

📌 Use identity $ \sin^2 x + \cos^2 x = 1 $, and:

📌 + (1 + \tan^2 x) + (1 + \cot^2 x) + 2(\tan x + \cot x) = 3 + \tan^2 x + \cot^2 x + 2(\tan x + \cot x)

📌 Let $ t = \tan x + \cot x $. Then:

📌 \tan^2 x + \cot^2 x = t^2 - 2

📌 + (t^2 - 2) + 2t = t^2 + 2t + 1 = (t + 1)^2

📌 Now, $ t = \tan x + \cot x \geq 2 $ by AM-GM, with equality when $ \tan x = 1 \Rightarrow x = \frac{\pi}{4} $.

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