1. It is known that $\sin y=2 \cos x+\frac{5}{2} \sin x, \cos y=2 \sin x+\frac{5}{2} \cos x$. Find $\sin 2 x$.
Answer. $\sin 2 x=-\frac{37}{40}$.

Solution. Squaring both equalities and adding them term by term, we get $1=4+20 \sin x \cos x+\frac{25}{4}$, from which $-10 \sin 2 x=\frac{37}{4}, \sin 2 x=-\frac{37}{40}$.