simplify using trigonometric identities



I have a horrible expression:

phi:
((c-1)*cos(x)*sin((c+1)*x)+(1-c)*sin(x)*cos((c+1)*x) 
+ (-c-1)*cos(x)*sin((c-1)*x)
+(-c-1)*sin(x)*cos((c-1)*x)) /(2*c^2-2);

which can be simplified to -sin(c*x)/(c^2-1)

I simplified it by hand using a number of  sin(a+b)=sin(a)cos(b)+sin(b)cos(a) 
and cos(a+b) rules using subst(). But can it be simplified in an automatic way 
without visual inspection? 

Best,
Nijso