Uniqueness principle theorem :If $f$ and $g$ are analytic functions on a domain $D$, and if $f(z)=g(z)$ for $z$ belonging to a set that has a non isolated point, then $f(z)=g(z)$ for all $z\in D$.
Why does this show all identities with entire functions holds true for those of complex functions? I guess I don't understand the phrase "belonging to a set that has a non isolated point." The definition for isolated point is given by:
Isolated Point : We say $z_0 \in E$ is an isolated point of the set $E$ if the is a $\rho>0$ such that $|z-z_0|\ge \rho$ $\forall z \in E$ other than $z_0$.
Edit: For example, since $\cos(z)$ and $\sin(z)$ are entire any identities with then such as $\sin(2x)=2\sin(x)\cos(x)$ hold in the complex case.
Consider the two functions $f(z) = \sin(2z)$ and $g(x) = 2\sin(z)\cos(z)$. These two entire functions agree on the real line. So by the identity property, they are equal everywhere. Does this make more sense?