Should cosecant be defined as $\csc \theta = \frac{1}{\sin \theta}$, specifying the constraint: $\sin \theta \neq 0$? I'm studying trigonometry on my own, and I keep noticing that the trigonometric functions are never defined with constraints to deal with divide-by-zero issues. As an example, I've seen cosecant defined like this: $\csc \theta = \frac{1}{\sin \theta}$ I've encountered this definition in the book I'm currently working through, and online resources 1 2 3, but as $\sin \theta$ has the range [-1, 1] shouldn't cosecant really be defined as: $\csc \theta = \frac{1}{\sin \theta}, \sin \theta \neq 0$? Am I missing something?

Those are the same thing. Either it's undefined as part of the definition (i.e. part of $\mathbb{R}$ is not in the domain) or it's undefined as a consequence of $\frac{1}{0}$ being undefined.

It's just like how $f(x)=\frac{1}{2-x}$ is completely clear without a note saying "but x can't be 2!" If a function is undefined, then the values for which it is undefined are not part of its domain by definition.