### Instruction: Simplify $\sin 17^\circ + \sin 43^\circ.$

Enter your answer as a trigonometric function evaluated at an integer, such as "sin 7".  (The system will assume that the angle is in degrees.  The angle should be positive and as small as possible.).### Response: To simplify this expression, I will use the sum-to-product identity for sine, which says that $\sin A + \sin B = 2 \sin \frac{A+B}{2} \cos \frac{A-B}{2}.$
Applying this identity to the given expression, I get $\sin 17^\circ + \sin 43^\circ = 2 \sin \frac{17^\circ + 43^\circ}{2} \cos \frac{17^\circ - 43^\circ}{2}.$
Simplifying the fractions inside the trigonometric functions, I get $2 \sin 30^\circ \cos (-13^\circ).$
Since cosine is an even function, I can drop the negative sign and write $2 \sin 30^\circ \cos 13^\circ.$
Finally, I recall that $\sin 30^\circ = \frac{1}{2}$, so I can simplify further and write $\sin 17^\circ + \sin 43^\circ = \cos 13^\circ.$