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Home » 2p + q = -6 \quad \text(Equation 8) - Sourci

2p + q = -6 \quad \text(Equation 8) - Sourci

📅 May 1, 2026 👤 scraface

May 01, 2026

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\cos^2 x + \sin^2 x + 2 + \sec^2 x + \csc^2 x + 2
Since $\cos^2 x + \sin^2 x = 1$, this simplifies to:
+ 2 + \sec^2 x + \csc^2 x + 2 = 5 + \sec^2 x + \csc^2 x
Now, write $\sec^2 x = 1 + \tan^2 x$ and $\csc^2 x = 1 + \cot^2 x$:
+ (1 + \tan^2 x) + (1 + \cot^2 x) = 7 + \tan^2 x + \cot^2 x
Let $y = \tan^2 x$, so $\cot^2 x = \frac{1}{y}$. Then the expression b...
+ y + \frac{1}{y}
We minimize $y + \frac{1}{y}$ for $y > 0$. By AM-GM inequality:
y + \frac{1}{y} \geq 2
Equality occurs when $y = 1$, i.e., $\tan x = 1 \Rightarrow x = \frac{...

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