of slopes of lines. For example, from the trigonometric identity
$$
\tan (\alpha-\beta)=\frac{\tan \alpha-\tan \beta}{1+\tan \alpha \tan \beta}
$$
it follows that two nonvertical lines with the same slope are parallel. Also, from the identity
$$
\cot (\alpha-\beta)=\frac{1+\tan \alpha \tan \beta}{\tan \alpha-\tan \beta}
$$
we find that two nonvertical lines with slopes having product -1 are perpendicular.