sec(6x+54)=csc(x+54), so:
cos(6x+54)=sin(x+54)=cos(90-x-54)=cos(36-x)=cos(x-36),
6x+54=x-36,
5x=-90, x=-18 is one solution. Check: sec(-54)=sec(54)=csc(36) (true).
6x+54=36-x,
7x=-18, x=-18/7.
For any angle A sinA=sin(180-A)=sin(360n+A); cos(A)=cos(-A)=cos(360n-A) where n is an integer.
So there are many solutions.
6x+54=360n+x-36, 5x=360n-90, x=72n-18 (x=54, 126, 198, 270, 342, etc.)
6x+54=360n+36-x,
7x=360n-18,
x=360n/7-18/7 (x=342/7, 702/7, 1062/7, etc.).
n can have negative as well as positive values.