## 6. Formulas for Adding Like Functions

Formulas for adding like trigonometric functions allow the transformation of the sum and difference of functions into the product of these functions. They have the following form:

$$
\begin{aligned}
& \sin \alpha+\sin \beta=2 \sin \frac{\alpha+\beta}{2} \cos \frac{\alpha-\beta}{2} \\
& \sin \alpha-\sin \beta=2 \sin \frac{\alpha-\beta}{2} \cos \frac{\alpha+\beta}{2} \\
& \cos \alpha+\cos \beta=2 \cos \frac{\alpha+\beta}{2} \cos \frac{\alpha-\beta}{2} \\
& \cos \alpha-\cos \beta=-2 \sin \frac{\alpha-\beta}{2} \sin \frac{\alpha+\beta}{2}
\end{aligned}
$$

447-458. Transform the following expressions into products: 447. $\sin 75^{\circ}+\sin 15^{\circ}$.
Solution. We will use formula (20):

$$
\begin{gathered}
\sin 75^{\circ}+\sin 15^{\circ}=2 \sin \frac{75^{\circ}+15^{\circ}}{2} \cos \frac{75^{\circ}-15^{\circ}}{2}=2 \sin 45^{\circ} \cos 30^{\circ}= \\
=2 \cdot \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2}=\frac{\sqrt{6}}{2}
\end{gathered}
$$