Trial and Error is not a proper method for proof. One can't prove using trial and error that ALL $\csc$ equations have the same asymptotes as the equation states... However, I have tried this equation and never failed.
warning, this may be very tough.
Here's the equation: for a $\csc$ function such as
$$ \csc(bx-c)+d $$
the asymptotes are $$x = \frac{π}{b}+\frac{c}{b}+\frac{2π}{b}n$$
Hint: What you effectively want is all $x$ such that $\sin(bx - c) = 0$. To get this, recall that $\sin \theta = 0$ if and only if $\theta = n\pi$ for some $n\in \mathbb{Z}$.