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Home » But we need integer — no solution? But problem asks for it. - Decision Point

But we need integer — no solution? But problem asks for it. - Decision Point

📅 May 2, 2026 👤 scraface

May 02, 2026

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📌 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}

📌 Let \(t = \sin 2x\), \(0 < t \leq 1\) (since \(x \in (0, \pi/2)\Rightarrow 2x \in (0, \pi)\), and \(t = 1\) at \(x = \pi/4\))

📌 Define \(f(t) = 1 + t + rac{4}{t^2}\), \(0 < t \leq 1\)

📌 Derivative: \(f'(t) = 1 - rac{8}{t^3}\)

📌 Set to zero: \(1 = rac{8}{t^3} \Rightarrow t^3 = 8 \Rightarrow t = 2\), not in domain. So maximum occurs at endpoint \(t = 1\)

📌 Then \(f(1) = 1 + 1 + 4 = 6\)

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