2. [4] Let $a \star b=\sin a \cos b$ for all real numbers $a$ and $b$. If $x$ and $y$ are real numbers such that $x \star y-y \star x=1$, what is the maximum value of $x \star y+y \star x$ ?
Answer: $\square$
We have $x \star y+y \star x=\sin x \cos y+\cos x \sin y=\sin (x+y) \leq 1$. Equality is achieved when $x=\frac{\pi}{2}$ and $y=0$. Indeed, for these values of $x$ and $y$, we have $x \star y-y \star x=\sin x \cos y-\cos x \sin y=\sin (x-y)=$ $\sin \frac{\pi}{2}=1$