A rectangle has sides of length $\sin x$ and $\cos x$ for some $x$. What is the largest possible area of such a rectangle?
Solution: We wish to maximize $\sin x \cdot \cos x=\frac{1}{2} \sin 2 x$. But $\sin 2 x \leq 1$, with equality holding for $x=\pi / 4$, so the maximum is $\frac{1}{2}$.