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homewood suites by hilton aurora naperville - inBeat

📅 May 4, 2026 👤 scraface

May 04, 2026

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📌 Notice that \(\sec \theta = \frac{1}{\cos \theta}\) and \(\csc \theta = \frac{1}{\sin \theta}\). Thus, the terms simplify to:

📌 \cos^2 \theta + 2 + \sec^2 \theta + \sin^2 \theta + 2 + \csc^2 \theta

📌 Using the Pythagorean identity \(\cos^2 \theta + \sin^2 \theta = 1\), the expression becomes:

📌 + 2 + \sec^2 \theta + 2 + \csc^2 \theta = 5 + \sec^2 \theta + \csc^2 \theta

📌 Recall the identities \(\sec^2 \theta = 1 + \tan^2 \theta\) and \(\csc^2 \theta = 1 + \cot^2 \theta\), so:

📌 + \sec^2 \theta + \csc^2 \theta = 7 + \tan^2 \theta + \cot^2 \theta

📌 Using the AM-GM inequality, \(\tan^2 \theta + \cot^2 \theta \geq 2\), hence:

📌 + \tan^2 \theta + \cot^2 \theta \geq 7 + 2 = 9

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