What's the answer to cscθ + cotθ ?
I keep getting (1+ cosθ)/ sinθ
you are correct.
You can test whether two trig expressions are equal by taking a non-standard angle, like 37°, and evaluating each expression
If the question was to "simplify", I would consider the original expression simplified enough.
Because of the many trig identities and relations, there are often multiple ways to express a trig expression.
e.g. in your answer we could replace 1 with sin^2Ø + cos^2Ø to have an "answer" of
(sin^2Ø + cosØ + cos^2Ø)/sinØ
which then becomes
sinØ +cotØ + cosØcotØ
but I certainly didn't simplify it, I just have an alternate form