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Home » \tan 30^\circ = \frac1{\sqrt3} - Aurero

\tan 30^\circ = \frac1{\sqrt3} - Aurero

📅 March 14, 2026 👤 scraface

Mar 14, 2026

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

📌 Solution: First, expand:

📌 Let \(u = \sin^2 x\), then \(\cos^2 x = 1 - u\), so:

📌 rac{u}{1 - u} + rac{1 - u}{u} + 2 = rac{u^2 + (1 - u)^2}{u(1 - u)} + 2 = rac{u^2 + 1 - 2u + u^2}{u(1 - u)} + 2 = rac{2u^2 - 2u + 1}{u(1 - u)} + 2

📌 But better: use identity:

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

📌 Now the full expression:

📌 + \sin 2x + rac{4}{\sin^2 2x}

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