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Home » But wait: $\frac916 = 0.5625$, but let’s verify total favorable: - Blask

But wait: $\frac916 = 0.5625$, but let’s verify total favorable: - Blask

📅 March 10, 2026 👤 scraface

Mar 10, 2026

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📌 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}

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

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