Question 1 As shown in Figure 1, find the maximum value of the area of the rectangle inscribed in the semicircle $O$ (the radius of the circle is 1).


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Parse: Connect $O A$.
Figure 1
Consider from the perspective of trigonometric functions. Let $\angle A O B=\theta$ $\left(0<\theta<\frac{\pi}{2}\right)$, then
$$|A B|=\sin \theta,|B C|=2|O B|=2 \cos \theta,$$

Thus, the area of rectangle $A B C D$ is
$$S=|A B| \cdot|B C|=2 \sin \theta \cos \theta=\sin 2 \theta$$

When $\theta=\frac{\pi}{4}$, $S_{\text {max }}=1$.
Consider from the perspective of algebra. Let $O B=x(0<x<1)$, then the area of rectangle $A B C D$ is $S=2 x \sqrt{1-x^{2}}$.

Applying the two-variable mean inequality $2 a b \leqslant a^{2}+b^{2}$, we get
$$S=2 x \sqrt{1-x^{2}} \leqslant x^{2}+\left(1-x^{2}\right)=1,$$

When $x=\sqrt{1-x^{2}}$, i.e., $x=\frac{\sqrt{2}}{2}$, $S_{\max }=1$.