Prove that the algebra does not vanish and that the algebra separates Show that the algebra generated by {sin(x),cos(x)} does not vanish on the
set [0, 2π].  Also, show that this algebra separates over this interval. 
So I know the definition of an algebra and what it means for it to not vanish for any x on interval.  However, I'm confused on how to set up this problem and this proof.  So do I put f(x)=sin(x)   and  g(x)=cos(x).  How do I go about proving that this algebra doesn't vanish on the interval [0, 2π] and that it separates over this interval? 
Any help would be amazing!

Since sin(x) $\in$ A, $sin^2(x)\in A$, similarly $cos^2(x)\in A$. Thus $sin^2(x) + cos^2(x) = 1 \in A$. $A$ vanishes at no point on $[0, 2\pi]$.
I am stuck on the second part..
