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Home » But mathematically, the ratio is **infinite** or **undefined**. - jntua results

But mathematically, the ratio is infinite or undefined. - jntua results

📅 March 9, 2026 👤 scraface

Mar 09, 2026

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📌 Find the minimum value of \((\cos x + \sec x)^2 + (\sin x + \csc x)^2\).

📌 To find the minimum value of the expression \((\cos x + \sec x)^2 + (\sin x + \csc x)^2\), let us start by rewriting \(\sec x\) and \(\csc x\) in terms of \(\cos x\) and \(\sin x\):

📌 Let \(a = \cos x\) and \(b = \sin x\). Then:

📌 This expands to:

📌 a^2 + 2 + \frac{1}{a^2} + b^2 + 2 + \frac{1}{b^2}

📌 Since \(a^2 + b^2 = \cos^2 x + \sin^2 x = 1\), the expression simplifies to:

📌 + 4 + \frac{1}{a^2} + \frac{1}{b^2} = 5 + \frac{1}{a^2} + \frac{1}{b^2}

📌 We apply the AM-GM inequality to \(\frac{1}{a^2} + \frac{1}{b^2}\):

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Speed = 60 miles per hour.
Time = 7.5 hours.
Distance = \( 60 \times 7.5 = 450 \).
Therefore, the car will travel 450 miles.
Let the three consecutive integers be \( x \), \( x+1 \), and \( x+2 \...
Set up the equation: \( x + (x+1) + (x+2) = 72 \).
Subtract 3 from both sides: \( 3x = 69 \).
Divide by 3: \( x = 23 \).
Therefore, the integers are 23, 24, and 25.
If the expression \( (3x - 4)(2x + 5) \) is expanded, what is the resu...

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