Prove that $\sin(z_{1}+z_{2})=\sin z_{1} \cos z_{2} + \cos z_{1} \sin z_{2}$
My solution
Let $z_{2}$ be a fixed real number. Then, $f(z)=\sin(z + z_{2})$ and $g(z)=\sin z\cos z_{2}+\cos z\sin z_{2}$ are two entire functions (of $z$) which agree for all real values $z=z_{1}$ and, hence, for all complex values $z=z_{1}$, as well.
Let $z=z_{1}$ be any such complex number. Then, $f(z)=\sin(z_{1}+z)$ and $g(z)=\sin z_{1}\cos z+\cos z_{1}\sin z$ agree for all real values $z=z_{2}$ and, hence,for all complex values $z=z_{2}$ as well.
Is my method correct?