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Home » You Won’t Believe How Shannon Hale Shocked Fans in Her Latest Masterpiece! - American Beagle Club

You Won’t Believe How Shannon Hale Shocked Fans in Her Latest Masterpiece! - American Beagle Club

📅 March 1, 2026 👤 scraface

Mar 01, 2026

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📌 Let \(a = \cos x\) and \(b = \sin x\). Then:

📌 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:

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