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Sum and Difference Identities

$\begin{equation*} sin(\alpha + \beta) = sin(\alpha) \cdot cos(\beta) + cos(\alpha) \cdot sin(\beta) \end{equation*}$

$\begin{equation*} cos(\alpha + \beta) = cos(\alpha) \cdot cos(\beta) - sin(\alpha) \cdot sin(\beta) \end{equation*}$

$\begin{equation*} tan(\alpha + \beta) = \frac{tan(\alpha) + tan(\beta)}{1 - tan(\alpha) \cdot tan(\beta)} \end{equation*}$

$\begin{equation*} sin(\alpha - \beta) = sin(\alpha) \cdot cos(\beta) - cos(\alpha) \cdot sin(\beta) \end{equation*}$

$\begin{equation*} cos(\alpha - \beta) = cos(\alpha) \cdot cos(\beta) + sin(\alpha) \cdot sin(\beta) \end{equation*}$

$\begin{equation*} tan(\alpha - \beta) = \frac{tan(\alpha) - tan(\beta)}{1 + tan(\alpha) \cdot tan(\beta)} \end{equation*}$

Example

$\begin{equation*} sin(15^\circ) \end{equation*}$

$\begin{equation*} \alpha = 45^\circ, \beta = 30^\circ, \alpha - \beta = 15^\circ \end{equation*}$

$\begin{equation*} sin(45^\circ - 30^\circ) = sin(45^\circ) \cdot cos(30^\circ) - sin(30^\circ) \cdot cos(45^\circ) \end{equation*}$

$\begin{equation*} =\left(\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2}\right) - \left(\frac{1}{2} \cdot \frac{\sqrt{2}}{2} \right) \end{equation*}$

$\begin{equation*} =\frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4} \end{equation*}$

$\begin{equation*} sin(15^\circ) = \boxed{\frac{\sqrt{6} - \sqrt{2}}{4}} \end{equation*}$