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Home » Probability for one non-prime: $\frac12$ - Londonproperty

Probability for one non-prime: $\frac12$ - Londonproperty

📅 March 13, 2026 👤 scraface

Mar 13, 2026

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📌 f(x) = \sec^2 x + \cos^2 x + 2 + \csc^2 x + \sin^2 x - 2

📌 f(x) = \sec^2 x + \csc^2 x + (\cos^2 x + \sin^2 x) + 2 - 2 = \sec^2 x + \csc^2 x + 1

📌 Now express in terms of sine and cosine:

📌 f(x) = \frac{1}{\cos^2 x} + \frac{1}{\sin^2 x} + 1 = \frac{\sin^2 x + \cos^2 x}{\sin^2 x \cos^2 x} + 1 = \frac{1}{\sin^2 x \cos^2 x} + 1

📌 Using the identity $ \sin(2x) = 2\sin x \cos x $, we have:

📌 \sin x \cos x = \frac{1}{2} \sin 2x \Rightarrow \sin^2 x \cos^2 x = \frac{1}{4} \sin^2 2x

📌 f(x) = \frac{1}{\frac{1}{4} \sin^2 2x} + 1 = \frac{4}{\sin^2 2x} + 1

📌 Since $ 0 < x < \frac{\pi}{2} $, then $ 0 < 2x < \pi $, and $ \sin 2x > 0 $. The minimum of $ f(x) $ occurs when $ \sin^2 2x $ is maximized, i.e., when $ \sin^2 2x = 1 $. This happens when $ 2x = \frac{\pi}{2} \Rightarrow x = \frac{\pi}{4} $.

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