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Home » 10 Ugly Characters Who Went Viral for Being Unforgettable! - AMAZONAWS

10 Ugly Characters Who Went Viral for Being Unforgettable! - AMAZONAWS

10 Ugly Characters Who Went Viral for Being Unforgettable

📅 March 11, 2026 👤 scraface

Mar 11, 2026

10 Ugly Characters Who Went Viral for Being Unforgettable!

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📌 Solution: Let $t = \sin x + \csc x$. Note $\tan x + \cot x = \frac{\sin x}{\cos x} + \frac{\cos x}{\sin x} = \frac{1}{\sin x \cos x}$. Let $y = \sin x \cos x = \frac{1}{2}\sin 2x$, so $\tan x + \cot x = \frac{2}{\sin 2x}$. The expression becomes $\left(\frac{2}{\sin 2x}\right)^2 + (\sin x + \csc x)^2$. Let $z = \sin x + \csc x \geq 2$ by AM-GM. Substitute $u = \sin 2x$, then minimize $\frac{4}{u^2} + \left(\frac{1}{\sin x} + \sin x\right)^2$. After calculus or substitution, the minimum occurs at $x = \frac{\pi}{4}$, yielding $(1 + 1)^2 + (\sqrt{2} + \frac{1}{\sqrt{2}})^2 = 4 + \left(\frac{3}{\sqrt{2}}\right)^2 = 4 + \frac{9}{2} = \frac{17}{2}$. However, correct simplification shows the minimum is $9$ when $x = \frac{\pi}{4}$.

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