Suppose alpha = arctan(1/2) and beta = arctan(1/3)
What is the value of tan(alpha + beta) ?
It's probably easiest to calculate sin alpha, cos alpha, sin beta and cos beta first:
If alpha = arctan(1/2), then 1/2 = tan alpha = sin alpha / cos alpha.
Multiplying through by 2cos alpha we get
cos alpha = 2 sin alpha
Squaring both sides and using sin^2 alpha + cos^2 alpha = 1 we get
4 sin^2 alpha = cos^2 alpha = 1 - sin^2 alpha
Adding sin^2 alpha to both sides and dividing by 5 we get
sin^2 alpha = 1/5
So sin alpha = 1/sqrt(5) and cos alpha = 2 sin alpha = 2/sqrt(5).
Similarly, we can find sin beta = 1/sqrt(10) and cos beta = 3/sqrt(10).
Now we can calculate
tan(alpha + beta) = sin(alpha+beta)/cos(alpha+beta)
=(sin alpha cos beta + sin beta cos alpha)/(cos alpha cos beta - sin alpha sin beta)
The numerator:
sin alpha cos beta + sin beta cos alpha
=(1/sqrt(5))(3/sqrt(10)) + (1/sqrt(10))(2/sqrt(5))
=5/sqrt(50) = 1/sqrt(2)
The denominator:
cos alpha cos beta - sin alpha sin beta
(2/sqrt(5))(3/sqrt(10)) - (1/sqrt(5))(1/sqrt(10))
=5/sqrt(50) = 1/sqrt(2)
So putting these together:
tan(alpha + beta) = (1/sqrt(2))/(1/sqrt(2)) = 1
So alpha + beta = pi/4
