to prove that sine is continuous we need to show that lim_{x-->}sin(x)=sin(a) for every real number a. if we let h=x-a, then x=a+h and x-->a <--> h-->0. so an equivalent statement is that

lim_{h-->0}sin(a+h) = sin(a)

use the theorem:

lim_{x-->0}cos(x) = 1 and lim_{x-->0}sin(x)=0

to show that this is true.

please help me with detailed steps!