By AM-GM, $t + \frac{1}{t} \geq 2$, with equality when $t = 1$ (i.e., $x = \frac{\pi}{4}$). Thus, the minimum is $7$, but we seek the maximum. As $x \to 0^+$ or $x \to \frac{\pi}{2}^-$, $\tan^2 x \to 0$ or $\infty$, so $t + \frac{1}{t} \to \infty$. However, the original expression is unbounded. Wait, this contradicts the problem's implication of a finite maximum. Re-examining:

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