2. Simplify the expression

$$
\frac{\sin 53^{\circ}-2 \cos 40^{\circ} \sin 13^{\circ}}{\cos 63^{\circ}}
$$
Solution. Using the trigonometric identity

$$
2 \sin \alpha \cos \beta=\sin (\alpha+\beta)+\sin (\alpha-\beta)
$$

we have

$$
\begin{aligned}
\frac{\sin 53^{\circ}-2 \cos 40^{\circ} \sin 13^{\circ}}{\cos 63^{\circ}} & =\frac{\sin 53^{\circ}-\left[\sin \left(13^{\circ}+40^{\circ}\right)+\sin \left(13^{\circ}-40^{\circ}\right)\right]}{\cos 63^{\circ}} \\
& =-\frac{\sin \left(-27^{\circ}\right)}{\sin 63^{\circ}}=\frac{\sin 27^{\circ}}{\sin 27^{\circ}}=1
\end{aligned}
$$