Question

how can i prove that sin(x) is always continuous ?

Answer 1

depends largely on your definition of sine
but you can do it like this
show \(\lim_{x\to 0}\sin(x+h)=\sin(x)\)that will prove that it is continuous

Answer 2

as for \(\sin(x+h)\)use the "addition angle formula
\[\sin(x+h)=\sin(x)\cos(h)+\cos(x)\sin(h)\] and take the limit as \(h\to 0\)

Answer 3

i understood your point of view but i meant that how did we know that the trigonmetric function is always continuous in all parts of their domain as the polynomials ?