Answer
The cosine of the difference of two angles is $\cos \left( \alpha -\beta \right)=\cos \alpha cos\beta +sin\alpha sin\beta $.
Work Step by Step
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 $.
