without using calculator,

find the Value of following is
cosec(-38888)

got me. can't think of a trick that will reduce that number (degrees or radians!) without using a calculator.

I can tell you right off that no trig function of a rational number of radians will come out to an easy value.

As for degrees, well -38888 is equivalent to -8, but I don't know a handy value for that.