Problem 11.4. It is known that $\frac{7}{13}+\sin \varphi=\cos \varphi$ for some real $\varphi$. What is $\sin 2 \varphi$?
Answer: $\frac{120}{169}$.

Solution. From the condition, it follows that $\frac{7}{13}=\cos \varphi-\sin \varphi$. Squaring this equality, we get $\frac{49}{169}=\cos ^{2} \varphi+\sin ^{2} \varphi-2 \sin \varphi \cos \varphi=1-\sin 2 \varphi$. Therefore, $\sin 2 \varphi=1-\frac{49}{169}=\frac{120}{169}$.