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john jay college of criminal justice - AssociationVoting

📅 May 18, 2026 👤 scraface

May 18, 2026

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📌 Question: Find the minimum value of $ (\cos x + \sec x)^2 + (\sin x + \csc x)^2 $ over $ x \in (0, \frac{\pi}{2}) $, arising in optimization of energy efficiency in a synthetic metabolic loop.

📌 Expand both terms:

📌 = \cos^2 x + 2\cos x \sec x + \sec^2 x + \sin^2 x + 2\sin x \csc x + \csc^2 x

📌 Since $ \cos x \sec x = 1 $, $ \sin x \csc x = 1 $, and $ \cos^2 x + \sin^2 x = 1 $, we get:

📌 = 1 + 2(1) + \sec^2 x + 2(1) + \csc^2 x = 1 + 2 + 2 + \sec^2 x + \csc^2 x = 5 + \sec^2 x + \csc^2 x

📌 Now write in terms of sine and cosine:

📌 \sec^2 x = \frac{1}{\cos^2 x}, \quad \csc^2 x = \frac{1}{\sin^2 x}

📌 \Rightarrow \text{Expression} = 5 + \frac{1}{\cos^2 x} + \frac{1}{\sin^2 x}

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