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Home » 5-You’re Earning Real Money Now: Don’t Miss the Payment That’s Changing EVERYTHING! - DNSFLEX

5-You’re Earning Real Money Now: Don’t Miss the Payment That’s Changing EVERYTHING! - DNSFLEX

📅 March 12, 2026 👤 scraface

Mar 12, 2026

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📌 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} $.

📌 f\left(\frac{\pi}{4}\right) = \frac{4}{1} + 1 = 5

📌 Thus, the minimum value is $ \boxed{5} $.

📌 Question: A mammalogist observes that the social cohesion index $ S $ of a primate group over time is modeled by $ S = \left| \sin\left(\frac{3\pi}{7} t\right) + \cos\left(\frac{4\pi}{7} t\right) \right| $, where $ t $ is time in days. How many times in one full cycle of $ S $ (i.e., over $ t \in [0, 7) $) does $ S = 1 $?

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