+ (1 + \tan^2 x) + (1 + \cot^2 x) = 7 + \tan^2 x + \cot^2 x
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- 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 $\csc^2 x = 1 + \cot^2 x$:
- Let $y = \tan^2 x$, so $\cot^2 x = \frac{1}{y}$. Then the expression becomes:
- + y + \frac{1}{y}
- We minimize $y + \frac{1}{y}$ for $y > 0$. By AM-GM inequality:
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