Identity with Maple from Rhysick

Hi everybody

I am doing this calculation:

eq1:=expand(tan(alpha-beta));
                           tan(alpha) - tan(beta) 
                          ------------------------
                          1 + tan(alpha) tan(beta)
eq2:=eval(eq1,{tan(alpha)=(n+1), tan(beta)=(n-1)});
                                      2         
                             -------------------
                             1 + (n + 1) (n - 1)
eq3:=normal(eq2);
                                     2 
                                     --
                                      2
                                     n 
alpha-beta=arctan(eq3);
                                               /2 \
                          alpha - beta = arctan|--|
                                               | 2|
                                               \n /
In Rhysick page 49 relation 9, it is stated that
arctan(2/n^2)=arctan(n+1)-arctan(n-1);
                       /2 \                                
                 arctan|--| = arctan(n + 1) - arctan(n - 1)
                       | 2|                                
                       \n /                                

How can I prouve that in Maple (or at least on paper)?

Thanks in advance

Mario Lemelin