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Home » Let width = \( w \), then length = \( 2w \). - Crosslake

Let width = \( w \), then length = \( 2w \). - Crosslake

📅 March 1, 2026 👤 scraface

Mar 01, 2026

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📌 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 $u$:

📌 \frac{1}{u^2} + \frac{2}{u} + 1 + 2u

📌 Define the function:

📌 f(u) = \frac{1}{u^2} + \frac{2}{u} + 1 + 2u, \quad 0 < u \leq \frac{1}{2}

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