Start with the definition of a function csc(x):
csc(x)=1/sin(x)
This immediately implies the domain this equation can have solutions to x!=pi*N (that's where sin(x)=0), where N - any integer number
Using the definition of csc(x), our equation is transformed into
9sin(x)=1/sin(x)
Multiplying by sin(x) both sides of equation we get
9sin^2(x)=1
sin^2(x)=1/9
sin(x)=+-1/3
Solutions for sin(x)=1/3:
x=arcsin(1/3)+2pi*N and, since sin(phi)=sin(pi-phi),
x=pi - arcsin(1/3)+2pi*N
Solutions for sin(x)=-1/3, considering sin(x) is an odd function, that is sin(-x)=-sin(x), is the same as for sin(-x)=1/3:
x=-arcsin(1/3)+2pi*N and, since sin(phi)=sin(pi-phi),
x=pi + arcsin(1/3)+2pi*N
We can combine all these solutions into one expression:
x=+-arcsin(1/3)+pi*N, where N - any integer number.
