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Home » Turn Ordinary Spaces Into Stunning Designs—Discover the Powerful Horizontal Line Line! - Crankk.io

Turn Ordinary Spaces Into Stunning Designs—Discover the Powerful Horizontal Line Line! - Crankk.io

📅 February 27, 2026 👤 scraface

Feb 27, 2026

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📝 Recent Posts

\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 +...
Solution: Expand the expression:
Simplify using $\sin^2 x + \cos^2 x = 1$ and $\csc^2 x = 1 + \cot^2 x$...
+ 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 become...
+ t + \frac{1}{t}.
By AM-GM, $t + \frac{1}{t} \geq 2$, with equality when $t = 1$ (i.e.,...
Actually, $\sin x + \csc x = \sin x + \frac{1}{\sin x} \geq 2$, and si...
But the sum is minimized at $8$ (when $x = \frac{\pi}{4}$), but maximu...

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