Furthermore,
As |OZ| = cos(α + β), and OZ is the adjacent side of triangle OYZ, |OZ| = |OY|cos α.
= cos α(cos β - (sin β sin α) / cos α).
Therefore = cos(α + β) = cos α cos β - sin α sin β

As tan(α + β) = sin(α + β) / cos(α + β),
tan(α + β) = (sin α cos β + sin β cos α) / (cos α cos β - sin α sin β)
Dividing through by cos α cos β,
tan(α + β) = ((sin α cos β / cos α cos β) + (sin β cos α / cos α cos β)) / ((cos α cos β / cos α cos β) - (sin α sin β / cos α cos β))
= ((sin α / cos α) + (sin β / cos β)) / (1 - (sin α sin β) / (cos α cos β))
so tan(α + β) = (tan α + tan β) / (1 - tan α tan β)
( The Mock TurtIe ™ --- Thinks you are a cunt, on, Sat 20 Jun 2009, 2:03, archived)