Finding the domain of $\frac{\sin x}{1+\cos x}$. (Am I correct?)

Question

I just had a test and was wondering if my answer to this domain question is correct.

Find domain of $y=\dfrac{\sin x}{1+\cos x}$

Of course the answer is $\{x \mid x\neq \pi + 2n\pi, n\text{ is an integer}\}$

But due to my obsession of double checking, I by accident changed the domain to $\{x \mid x\neq -\pi + 2n\pi, n\text{ is an integer}\}$$\ldots$

Is this still correct? Cosine of $-\pi$ or $-180^\circ$ is still $-1$ which would make the denominator $0$, satisfying the domain I wrote the second time for my final answer. Of course the point of adding $2n\pi$ is to show that it has coterminal angles and such, and its best to add it to the domain of the angle in standard position.

Anyways, how likely would my changed answer get full marks, considering work was shown. :( rip why do I always do this!!!!!!

Answer 1

If $$\pi+2n\pi=-\pi+2m\pi$$

$1+2n=2m-1\iff n=m-1$

So, both domains are actually identical

Answer 2

If you see that your domain can be also written as the union of open intervals $$](2k+1)\pi,(2k+3)\pi[$$ for all $k\in\mathbb Z$ you can maybe be sure that you are correct (with the condition that your "$n$ is an integer" means rational integer, i. e. positive or negative or zero).