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Home » \[ = 20 - 9 + 2 = 13 \] - Blask

\[ = 20 - 9 + 2 = 13 \] - Blask

📅 March 10, 2026 👤 scraface

Mar 10, 2026

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📌 5Question: Find the minimum value of $(\cos x + \sec x)^2 + (\sin x + \csc x)^2$ for $0 < x < \frac{\pi}{2}$.

📌 We begin by expanding the expression:

📌 Recall that $\cos x \sec x = 1$ and $\sin x \csc x = 1$, so the expression becomes:

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

📌 Since $\cos^2 x + \sin^2 x = 1$, this simplifies to:

📌 + 2 + \sec^2 x + \csc^2 x + 2 = 5 + \sec^2 x + \csc^2 x

📌 Now, write $\sec^2 x = 1 + \tan^2 x$ and $\csc^2 x = 1 + \cot^2 x$:

📌 + (1 + \tan^2 x) + (1 + \cot^2 x) = 7 + \tan^2 x + \cot^2 x

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