10. It is known that $\frac{7}{13}+\sin \phi=\cos \phi$ for some real $\phi$. What is $\sin 2 \phi$ ? Answer: $\frac{120}{169}$.
Solution. By squaring the equation $\frac{7}{13}=\cos \phi-\sin \phi$ we get
$$
\frac{49}{169}=\cos ^{2} \phi+\sin ^{2} \phi-2 \sin \phi \cos \phi=1-\sin 2 \phi .
$$

So $\sin 2 \phi=1-\frac{49}{169}=\frac{120}{169}$.