In this thread you are supposed to prove that sine is continuous. Cosine can be expressed through sine. Or use the same method as for sine.
Prove that sin(x) and cos(x) are continuous at x= 0, first. Then use the fact that cos(x+y)= cos(x)cos(y)- sin(x)sin(y). How you prove that sin(x) and cos(x) are continuous at x= 0 depends on precisely how you have defined sine and cosine.