tan(z+w)=cos(z+w)sin(z+w)
Wegen:
sin(z+w)=sin(z)∗cos(w)+cos(z)∗sin(w)
cos(z+w)=cos(z)∗cos(w)−sin(z)∗sin(w)
folgt:
tan(z+w)=cos(z)∗cos(w)−sin(z)∗sin(w)sin(z)∗cos(w)+cos(z)∗sin(w)
Wegen cos(w) = sin(w)/tan(w) folgt
tan(z+w)=tan(z)sin(z)∗tan(w)sin(w)−sin(z)∗sin(w)sin(z)∗tan(w)sin(w)+tan(z)sin(z)∗sin(w)
tan(z+w)=tan(z)∗tan(w)sin(z)∗sin(w)−sin(z)∗sin(w)tan(w)sin(z)∗sin(w)+tan(z)sin(z)∗sin(w)
tan(z+w)=sin(z)∗sin(w)∗tan(z)∗tan(w)1−1sin(z)∗sin(w)∗tan(w)1+tan(z)1
tan(z+w)=tan(z)∗tan(w)1−1tan(w)1+tan(z)1
tan(z+w)=tan(z)∗tan(w)1−tan(z)∗tan(w)tan(z)tan(w)tan(w)+tan(z)
tan(z+w)=1−tan(z)∗tan(w)tan(w)+tan(z)