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]
