You can think of this as an equality of functions. It is saying that $\sin{2(-)}=2\sin{(-)}\cos{(-)}$, where the argument of the function goes where the $(-)$ is. You get the usual identity by evaluating these functions on $\alpha$ (whatever $\alpha$ is), but you can put any expression you like in there, such as $n\alpha$, $\frac{1}{\alpha}$, $e^\alpha$, etc. As long as everything is properly defined (for example if you substitute in $\frac{1}{\alpha}$ then $0$ is no longer a permissible value), you'll always get a true identity.