Theta is an angle between the lines L1 and L2 with the slopes m1 and m2. Prove that tan theta= (m2 - m1)/(1+m2m1).

Answer 1

tan slope angle 1 = m1
tan slope angle 2 = m2
so
angle 1 = tan^-1 m1
angle 2 = tan^-1 m2

so
theta = angle 2 - angle 1
= tan^-1 m2 - tan^-1 m1

tan theta = tan (a - b)
where a =tan^-1 m2 and b =tan^-1 m1
so
tan theta = (tan a-tan b)/(1+tan a tan b) [trig identity]

= (m2 -m1)/(1 + m2 m1)