[newbie] trying to simplify goniometric expression

I defined the following function:
phicalp(m,omega):=-sum(atan(omega/(4*(cos((k*%PI)/(2*m+1)))^2)),k,1,m);

I expected that the following "test-expression" should give me "-omega" as 
a result:
ratsimp(tan(phicalp(1,omega))),trigexpand=super;

however maxima doesn't seem to simplify the denominator to one:
                                       omega
(%o2)                            - -----------
                                         2 %PI
                                    4 cos (---)
                                            3

Which is the correct approach for this kind of simplification?

thanks in advance
hugo