## Precalculus (6th Edition) Blitzer

The cosine of the difference of two angles is $\cos \left( \alpha -\beta \right)=\cos \alpha cos\beta +sin\alpha sin\beta$.
Let us assume that a point A $\left( \cos \alpha ,\sin \alpha \right)$ subtended at angle $\alpha$ moves toward point B $\left( \cos \beta ,\sin \beta \right)$ subtended at angle $\beta$ in a rectangular coordinate system. The shortest distance between points A and B by using the distance formula is \begin{align} & AB=\sqrt{{{\left( \cos \alpha -cos\beta \right)}^{2}}+{{\left( \sin \alpha -\sin \beta \right)}^{2}}} \\ & =\sqrt{{{\cos }^{2}}\alpha +{{\cos }^{2}}\beta -2\cos \alpha \cos \beta +si{{n}^{2}}\alpha +si{{n}^{2}}\beta -2sin\alpha sin\beta } \\ & =\sqrt{\left( {{\cos }^{2}}\alpha +si{{n}^{2}}\alpha \right)+\left( {{\cos }^{2}}\beta +si{{n}^{2}}\beta \right)-2\left( \cos \alpha \cos \beta +sin\alpha sin\beta \right)} \\ & =\sqrt{1+1-2\cos \left( \alpha -\beta \right)} \end{align} Now, simplify as follows: \begin{align} & \sqrt{1+1-2\cos \left( \alpha -\beta \right)}=\sqrt{1+1-2\cos \alpha cos\beta -2sin\alpha sin\beta } \\ & 2-2\cos \left( \alpha -\beta \right)=2-2\cos \alpha cos\beta -2sin\alpha sin\beta \\ & -2\cos \left( \alpha -\beta \right)=-2\left( \cos \alpha cos\beta +sin\alpha sin\beta \right) \\ & \cos \left( \alpha -\beta \right)=\cos \alpha cos\beta +sin\alpha sin\beta \end{align} Thus, the cosine of the difference of two angles is $\cos \left( \alpha -\beta \right)=\cos \alpha cos\beta +sin\alpha sin\beta$.