On page 498 of the paper "THE COMBINATORIAL STRUCTURE OF TRIGONOMETRY" by ADEL F. ANTIPPA there are identities for trig functions of sums of n different angles.
http://www.emis.de/journals/HOA/IJMMS/2003/8475.pdf
Can you see a use for these identities? Perhaps some integral substitution like the more familiar identities are used for.

