It follows from cot(x) = 2 that cos(x) = 2sin(x). So
5sin(x)cos(y) + 8sin(x)sin(y) = 0
sin(x)(5cos(y) + 8sin(y)) = 0
5cos(y) + 8sin(y) = 0 . . . . . because sin(x) is not zero
5cos(y) = -8sin(y) . . . . . if sin(y) were zero then cos(y) would be too; so sin(y) is not zero
cos(y)/sin(y) = -8/5
cot(y) = -8/5