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

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yes, take $\alpha= 5\theta$ – Belgi Sep 13 '12 at 9:38
up vote 4 down vote accepted

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.

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Of course yes. If you like, you can also make a simple substitution to see it more clearly.
For example, your second example:

Let $\lambda=5\alpha$: then, $\sin 10\alpha = \sin 2\lambda = 2\sin\lambda\cos\lambda = 2\sin 5\alpha\cos 5\alpha$.

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