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what's the difference between seltzer and sparkling water - NBX Soluciones

📅 May 3, 2026 👤 scraface

May 03, 2026

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📌 \sec^2 x + \csc^2 x + 2\sec x \csc x + 1 + 2\sin x \cos x

📌 Recall that $\sec^2 x = 1 + \tan^2 x$, $\csc^2 x = 1 + \cot^2 x$, but it's more useful to write in terms of $\sin x$ and $\cos x$:

📌 \sec^2 x + \csc^2 x = \frac{1}{\cos^2 x} + \frac{1}{\sin^2 x} = \frac{\sin^2 x + \cos^2 x}{\sin^2 x \cos^2 x} = \frac{1}{\sin^2 x \cos^2 x}

📌 Also, $\sec x \csc x = \frac{1}{\sin x \cos x}$, so:

📌 2\sec x \csc x = \frac{2}{\sin x \cos x}

📌 And $2\sin x \cos x$ remains.

📌 Let $u = \sin x \cos x$. Since $x \in (0, \frac{\pi}{2})$, $u > 0$, and $u = \frac{1}{2} \sin 2x$, so $0 < u \leq \frac{1}{2}$.

📌 Now rewrite the entire expression in terms of $u$:

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