(Too long for a comment.)

It's a bit older than your reference, but so-called "hypergoniometric functions" have been considered by [Erik Lundberg in 1879](http://www.maths.lth.se/matematiklu/personal/jaak/hypergf.ps). [This article](http://www.jstor.org/pss/2695794) is a more recent discussion. [Shelupsky](http://www.jstor.org/pss/2309789) and [Burgoyne](http://dx.doi.org/10.1090/S0025-5718-1964-0164066-5) discuss similar generalizations. All ultimately consider this as the problem of inverting an appropriate generalization of the integral representations of arcsine and arccosine.

The $n=3$ case has been considered separately by [A.C. Dixon](http://gdz.sub.uni-goettingen.de/de/dms/load/img/?PPN=PPN600494829_0024&PHYSID=PHYS_0179); I had talked a bit about Dixon elliptic functions in [this math.SE answer](http://math.stackexchange.com/a/36506).