trigonometry - $sin 4alpha = 2sin 2alpha times cos 2alpha $?

A trigonometry rule says that $\sin 2\alpha = 2\sin \alpha \times \cos \alpha$. Does this also apply to $\sin 2x$ when $x = n \times \alpha$?

For example:
$$\sin 4\alpha = 2\sin 2\alpha \times \cos 2\alpha$$
$$\sin 10\alpha = 2\sin 5\alpha \times \cos 5\alpha$$

Answer

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.