Simplfy trigonometric expression with conditions

Questions
1

I would like to write a trig expression free of fractions. I looked at Assumptions, but I’m not sure it will give the output I want.

Simplify Trig Expressions Examples NB.nb (5.96 KB)

2

The expression in the notebook:
\large{\frac{\cos (x) \csc ^2(x)-\cos (x)}{\cos (x) \csc (x)+\cos (x)}}

You could use TrigFactor followed by FullSimplify

expr = (Cos[x] Csc[x]^2 - Cos[x])/(Cos[x] Csc[x] + Cos[x]);

expr // TrigFactor // FullSimplify
(* Out: -1 + Csc[x] *)

Or just TrigReduce

In[23]:= TrigReduce[expr]
Out[23]= -1 + Csc[x]
3

Thank you, Hans!

Your response has helped me greatly reduce the amount of code I used and can be applied to a huge variety of trig expressions.