# How do you simplify sin(alpha+beta)+sin(alpha-beta)?

Sep 2, 2016

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

#### Explanation:

We use the general property
$\sin \left(a + b\right) = \sin \left(a\right) \cos \left(b\right) + \sin \left(b\right) \cos \left(a\right)$

So, simplifying the above expression using the property, we get;
$\sin \left(\alpha + \beta\right) + \sin \left(\alpha - \beta\right) = \sin \left(\alpha\right) \cos \left(\beta\right) + \textcolor{red}{\sin \left(\beta\right) \cos \left(\alpha\right)} + \sin \left(\alpha\right) \cos \left(\beta\right) - \textcolor{red}{\sin \left(\beta\right) \cos \left(\alpha\right)}$
$\therefore$ $\sin \left(\alpha + \beta\right) + \sin \left(\alpha - \beta\right) = 2 \cdot \sin \left(\alpha\right) \cos \left(\beta\right)$
as the two terms in red get cancelled