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Home » Yes, You Can Host a Taco Party That Steals the Spotlight—Watch How! - Upplift

Yes, You Can Host a Taco Party That Steals the Spotlight—Watch How! - Upplift

📅 February 28, 2026 👤 scraface

Feb 28, 2026

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📌 Rearranging:

📌 x^2 - 2x + y^2 + z^2 = 0 \implies (x^2 - 2x + 1) + y^2 + z^2 = 1 \implies (x - 1)^2 + y^2 + z^2 = 1.

📌 This is a sphere centered at $(1, 0, 0)$ with radius 1.

📌 \boxed{\text{A sphere of radius 1 centered at } (1, 0, 0)}

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

📌 Solution: Expand the expression:

📌 Simplify using $\sin^2 x + \cos^2 x = 1$ and $\csc^2 x = 1 + \cot^2 x$, $\sec^2 x = 1 + \tan^2 x$:

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

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