Below I cited a passage from Apostol's Calculus. I don't understand how to use the identity to show that two lines with equal slopes are parallel.
Concepts such as perpendicularity and parallelism can be explained rather simply in analytic terms making use 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.
When the two nonvertical lines have the same slope, then
(1) they have the same angle of inclination (assuming that one is α and the other is β).
---- This means $\alpha = \beta$ and hence they are parallel [corresponding angles].
---- Also α = β implies tan (α - β) = 0 [the LHS]
(2) same slope implies tan α = tan β.
---- Thus , tan α - tan β = 0 [the RHS]