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our deepest fear is not that we are inadequate - Groen Casting

📅 March 1, 2026 👤 scraface

Mar 01, 2026

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

📌 Let $y = \tan^2 x$, so $\cot^2 x = \frac{1}{y}$. Then the expression becomes:

📌 + y + \frac{1}{y}

📌 We minimize $y + \frac{1}{y}$ for $y > 0$. By AM-GM inequality:

📌 y + \frac{1}{y} \geq 2

📌 Equality occurs when $y = 1$, i.e., $\tan x = 1 \Rightarrow x = \frac{\pi}{4}$.

📌 Thus, the minimum value is:

📌 + 2 = \boxed{9}

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