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Home » Stroll Out at Midnight… You’ll Never Guess What Mysteries Wait in the Dark! - Crosslake

Stroll Out at Midnight… You’ll Never Guess What Mysteries Wait in the Dark! - Crosslake

📅 March 1, 2026 👤 scraface

Mar 01, 2026

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\sin^2 x = rac{1 - \cos 2x}{2}, \quad \cos^2(2x) = rac{1 + \cos 4x}{...
f(x) = rac{1 - \cos 2x}{2} + rac{1 + \cos 4x}{2} = rac{1}{2} - rac...
f(x) = 1 + rac{1}{2}(\cos 4x - \cos 2x)
Now, use the identity for $ \cos 4x - \cos 2x $:
\cos A - \cos B = -2 \sin\left(rac{A+B}{2}
ight)\sin\left(rac{A-B}{2}
Let $ A = 4x $, $ B = 2x $:
\cos 4x - \cos 2x = -2 \sin(3x)\sin(x)
f(x) = 1 + rac{1}{2}(-2 \sin 3x \sin x) = 1 - \sin 3x \sin x
We now seek the minimum value of $ f(x) = 1 - \sin 3x \sin x $. Since...

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