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$\sec x + \csc x \approx 11.025$, square $\approx 121.55$
$\sin x + \cos x \approx 1.0948$, square $\approx 1.20$
Sum $\approx 122.75$, which is much larger than 10.
So indeed, as $x \to 0^+$ or $x \to \frac{\pi}{2}^-$, the expression grows without bound.
But is that correct? Let's analyze:
As $x \to 0^+$, $\sec x \to 1$, $\csc x \to \infty$, so $(\sec x + \csc x)^2 \sim \csc^2 x \sim \frac{1}{x^2}$
And $\sin x + \cos x \to 1$, so its square is constant, but the first term dominates.
Thus, $(\sec x + \csc x)^2 \to \infty$, so the entire expression tends to infinity.
But the problem asks for the **maximum** — if it diverges, there is no finite maximum.
But likely we made a mistake in interpretation — perhaps the expression is bounded?