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Home » El perímetro \( P \) es: - Blask

El perímetro \( P \) es: - Blask

📅 March 9, 2026 👤 scraface

Mar 09, 2026

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📌 Question: Find the minimum value of \((\cos x + \sec x)^2 + (\sin x + \csc x)^2\) for \(0 < x < rac{\pi}{2}\).

📌 Solution: Expand both squares:

📌 \cos^2 x + \sin^2 x + 4 + \sec^2 x + \csc^2 x = 1 + 4 + rac{1}{\cos^2 x} + rac{1}{\sin^2 x}

📌 So total expression is:

📌 + \sec^2 x + \csc^2 x = 5 + rac{1}{\cos^2 x} + rac{1}{\sin^2 x}

📌 + rac{\sin^2 x + \cos^2 x}{\sin^2 x \cos^2 x} = 5 + rac{1}{\sin^2 x \cos^2 x}

📌 But \(\sin^2 x \cos^2 x = rac{1}{4} \sin^2 2x\), so:

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