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Home » \dbinom125 = \dbinom127, \quad \dbinom124 = \dbinom128 - Crosslake

\dbinom125 = \dbinom127, \quad \dbinom124 = \dbinom128 - Crosslake

📅 March 1, 2026 👤 scraface

Mar 01, 2026

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📌 f(x) = \sin^2 x + \cos^2(2x)

📌 We use trigonometric identities to simplify and analyze the expression. Recall:

📌 \sin^2 x = rac{1 - \cos 2x}{2}, \quad \cos^2(2x) = rac{1 + \cos 4x}{2}

📌 f(x) = rac{1 - \cos 2x}{2} + rac{1 + \cos 4x}{2} = rac{1}{2} - rac{\cos 2x}{2} + rac{1}{2} + rac{\cos 4x}{2} = 1 - rac{\cos 2x}{2} + rac{\cos 4x}{2}

📌 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}

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