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Home » $ 9 \equiv 1 \pmod4 $ - Londonproperty

$ 9 \equiv 1 \pmod4 $ - Londonproperty

📅 March 14, 2026 👤 scraface

Mar 14, 2026

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📌 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 $ -1 \leq \sin 3x \sin x \leq 1 $, the maximum of $ \sin 3x \sin x $ is 1, so the minimum of $ f(x) $ is:

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