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Home » What Does FT Mean? You Won’t Believe the Shocking Truth Behind This Abbreviation! - Sourci

What Does FT Mean? You Won’t Believe the Shocking Truth Behind This Abbreviation! - Sourci

📅 May 2, 2026 👤 scraface

May 02, 2026

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📌 + 2 + 2 + \cot^2 x + \tan^2 x = 5 + \cot^2 x + \tan^2 x.

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

📌 + t + \frac{1}{t}.

📌 By AM-GM, $t + \frac{1}{t} \geq 2$, with equality when $t = 1$ (i.e., $x = \frac{\pi}{4}$). Thus, the minimum is $7$, but we seek the maximum. As $x \to 0^+$ or $x \to \frac{\pi}{2}^-$, $\tan^2 x \to 0$ or $\infty$, so $t + \frac{1}{t} \to \infty$. However, the original expression is unbounded. Wait, this contradicts the problem's implication of a finite maximum. Re-examining:

📌 Actually, $\sin x + \csc x = \sin x + \frac{1}{\sin x} \geq 2$, and similarly for $\cos x + \sec x$. But squaring gives:

📌 But the sum is minimized at $8$ (when $x = \frac{\pi}{4}$), but maximum is unbounded. However, the problem likely seeks the minimum. Assuming a typo, if the question is to find the minimum:

📌 At $x = \frac{\pi}{4}$, $\sin x = \cos x = \frac{\sqrt{2}}{2}$, $\csc x = \sec x = \sqrt{2}$. Then:

📌 \left(\frac{\sqrt{2}}{2} + \sqrt{2}\right)^2 + \left(\frac{\sqrt{2}}{2} + \sqrt{2}\right)^2 = 2 \cdot \left(\frac{3\sqrt{2}}{2}\right)^2 = 2 \cdot \frac{18}{4} = 9.

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