I'm trying to prove that csc theta minus cot theta equals sin theta/1+cos theta, but I get 1-cos theta/sin theta. Where am I going wrong?

I do not know where you're going wrong. You have not shown what you tried.

csc(x)-cot(x)

=\frac{1}{sin(x)}-\frac{cos(x)}{sin(x)}

=\frac{sin(x)-sin(x)cos(x)}{sin^{2}(x)}

=\frac{sin(x)(1-cos(x))}{sin^{2}(x)}

=\frac{1-cos(x)}{sin(x)}

Now, multiply top and bottom by
1+cos(x) and you should see it fall into place.