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Home » Final decision: use correct arithmetic with consistent logic. - Nurtured Nest

Final decision: use correct arithmetic with consistent logic. - Nurtured Nest

📅 May 8, 2026 👤 scraface

May 08, 2026

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📌 This expands to:

📌 a^2 + 2 + \frac{1}{a^2} + b^2 + 2 + \frac{1}{b^2}

📌 Since \(a^2 + b^2 = \cos^2 x + \sin^2 x = 1\), the expression simplifies to:

📌 + 4 + \frac{1}{a^2} + \frac{1}{b^2} = 5 + \frac{1}{a^2} + \frac{1}{b^2}

📌 We apply the AM-GM inequality to \(\frac{1}{a^2} + \frac{1}{b^2}\):

📌 \frac{1}{a^2} + \frac{1}{b^2} \geq \frac{2}{ab}

📌 Given \(a^2 + b^2 = 1\), the maximum of \(ab\) is \(\frac{1}{2}\) when \(a = b = \frac{\sqrt{2}}{2}\). Then:

📌 ab = \frac{1}{2} \quad \Rightarrow \quad \frac{1}{a^2} + \frac{1}{b^2} = \frac{(a^2 + b^2)^2}{a^2 b^2} = \frac{1}{(\frac{1}{2})^2} = 4

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