What is the difference between $2\sin(x)$ and $\sin(2x)$? In this question, I saw that there was no difference between $\sin^2(x)$ and $\sin(x)^2$. 
However, I am curious what's the difference between $2\sin(x)$ and $\sin(2x)$?
 A: $2\sin(x)$ is multiplying the $\sin$ function by 2. Whereas the $\sin(2x)$ is inputting a value of twice $x$ into the function. Note that $\sin(2x) = 2\sin(x)\cos(x) \neq 2\sin(x)$
A: I think that aside from that fact that
$$\sin 2x = 2\sin x\cos x,$$
the difference should be patently clear once you have the graphs.

Notice that $2\sin x$ means take the value $\sin x$ and double it. Where as $\sin 2x$ means take the value $x$, double it, and then apply the $\sin $ function to that value.
Also, I do not believe it is universally accepted that
$$\sin^2 (x) = \sin (x)^2,$$
since $\sin (x)^2$ can be interpreted as 
$$\sin ((x)^2).$$
For a less ambiguous case, take for example $\sin (2xy)^2$
is can easily be interpreted as 
$$\sin 4x^2y^2$$
or 
$$\sin^2 (2xy).$$
So I believe there is a difference. 
For the sake of clarity, $\sin (x)^2$ should really be written as
$$(\sin x)^2$$
if you mean to convey $\sin^2 x$.
A: $2\sin x$ means first you take the sine of $x$.  Then you multiply the result by 2.
$\sin 2x$ means first you multiply $x$ by 2.  Then you take the sine of the result.
$\sin 2x$ can, and sometimes should, also be written as $\sin(2x)$.
It is not ok to pull constant multiples out of trig functions.
A: Really,  $\sin(x) ^2 = \sin^2(x)$ only because we adopted the function notation used prior,  namely $f(x) ^2 = f^2(x)$. In most cases I still use the former to avoid confusing differentiation or iteration,  but trig functions  still use the latter. Now,  this notation only holds for powers...  When you say $2\sin(x)$ you are making a sine wave with an amplitude of $2$. When you say $\sin(2x)$ you are talking about a sine wave with twice the frequency of the standard sine wave. 
A: $\sin(2x)=2\sin(x)\cos(x)$ its double angle formula
