Interesting question regarding the solution to a trigonometric equation

I just saw a very simple question that has generated a lot of interesting discussion within our department.

Our trigonometry teacher put the question on the board:

Solve: tan(x) = 1/csc(x)

Naturally, all the students (and the teacher as well), applied the identity 1/csc(x) = sin(x) and proceeded to say that the solutions were x = (pi)*n, for all integer n.

However, upon looking at the domain, since csc(x) is undefined for all x = (pi)*n, it would appear that x = (pi)*n is an extraneous root, making the solution the empty set.

So, is the solution x = (pi)*n or the empty set?

Thanks in advance!

Re: Interesting question regarding the solution to a trigonometric equation