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Home » Thus, the total number of arrangements is \(\boxed241920\). - Network4

Thus, the total number of arrangements is \(\boxed241920\). - Network4

📅 May 20, 2026 👤 scraface

May 20, 2026

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📌 \tan x + \cot x = 2\sqrt{2}.

📌 Recall that $ \tan x = \frac{\sin x}{\cos x} $, $ \cot x = \frac{\cos x}{\sin x} $, so:

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

📌 \frac{1}{\sin x \cos x} = 2\sqrt{2}.

📌 Use the double-angle identity: $ \sin(2x) = 2\sin x \cos x $, so:

📌 \sin x \cos x = \frac{1}{2} \sin(2x).

📌 \frac{1}{\frac{1}{2} \sin(2x)} = 2\sqrt{2} \Rightarrow \frac{2}{\sin(2x)} = 2\sqrt{2} \Rightarrow \frac{1}{\sin(2x)} = \sqrt{2} \Rightarrow \sin(2x) = \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2}.

📌 Now solve $ \sin(2x) = \frac{\sqrt{2}}{2} $ for $ x \in (0, \pi) $. Then $ 2x \in (0, 2\pi) $.

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