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Home » You Won’t Believe How Shannon Hale Shocked Fans in Her Latest Masterpiece! - 500apps

You Won’t Believe How Shannon Hale Shocked Fans in Her Latest Masterpiece! - 500apps

📅 March 15, 2026 👤 scraface

Mar 15, 2026

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📌 Thus, the minimum value is $ \boxed{0} $.

📌 Question: Find the range of the function $ f(x) = \frac{1}{\sin^2 x + 4\cos^2 x} $ as $ x $ ranges over all real numbers such that $ \sin^2 x + \cos^2 x = 1 $.

📌 Solution: Since $ \sin^2 x + \cos^2 x = 1 $, we write $ f(x) = \frac{1}{\sin^2 x + 4\cos^2 x} = \frac{1}{1 + 3\cos^2 x} $. Let $ y = \cos^2 x $, so $ 0 \leq y \leq 1 $. Then $ f(x) = \frac{1}{1 + 3y} $. As $ y $ varies from 0 to 1, $ 1 + 3y $ varies from 1 to 4, so $ f(x) $ varies from $ \frac{1}{4} $ to $ 1 $. Hence, the range of $ f(x) $ is $ \left[\frac{1}{4}, 1\right] $. The final answer is $ \boxed{\left[\frac{1}{4}, 1\right]} $.

📌 Question: Find all angles $ z \in [0^\circ, 360^\circ] $ such that $ \tan z + \cot z = 2 $.

📌 Solution: Start with $ \tan z + \cot z = 2 $. Recall $ \tan z = \frac{\sin z}{\cos z} $, $ \cot z = \frac{\cos z}{\sin z} $. Then:

📌 \tan z + \cot z = \frac{\sin z}{\cos z} + \frac{\cos z}{\sin z} = \frac{\sin^2 z + \cos^2 z}{\sin z \cos z} = \frac{1}{\sin z \cos z}

📌 Set $ \frac{1}{\sin z \cos z} = 2 $. Then $ \sin z \cos z = \frac{1}{2} $. Using the identity $ \sin(2z) = 2\sin z \cos z $, we get $ \sin(2z) = 1 $. Solve $ \sin(2z) = 1 $:

📌 2z = 90^\circ + 360^\circ n \Rightarrow z = 45^\circ + 180^\circ n

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