# Prove $\sin x=2\sin\frac{x}{2}\cos\frac{x}{2}$

How would I prove the following identity?

$$\sin x=2\sin\frac{x}{2}\cos\frac{x}{2}$$

I know $$\sin(a+b)=\sin a\cos b+\sin a \cos b.$$

So I did

$$\sin\frac{x}{2}\cos\frac{x}{2}+\cos\frac{x}{2}\sin\frac{x}{2}.$$

But what technique would I have to use to continue the problem?

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$\sin x = \sin (\frac{1}{2} x + \frac{1}{2}x) =$ ? – user17794 Jul 28 '12 at 16:47
Apart from a typo, you are $99\%$ finished: $a+a=2a$. – André Nicolas Jul 28 '12 at 16:49
Please do not write 1/2x in text, as it is not clear whether you want $\frac 1{2x}$ or $\frac x2$. Either write x/2, (1/2)x, or, better, use $\LaTeX$ – Ross Millikan Jul 28 '12 at 16:51
Is there a tutorial for this latex I think someone posted a link once.... – Fernando Martinez Jul 28 '12 at 16:54

Since $\sin(a+b) = \sin a \cos b + \cos a \sin b$, letting $a = b = \frac{x}{2}$ gives $\sin x = \sin \frac{x}{2} \cos \frac{x}{2} + \cos \frac{x}{2} \sin \frac{x}{2} = 2 (\sin \frac{x}{2} \cos \frac{x}{2})$.

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Hmm would 2(sin(x/2)cos(x/2) equal sinX since 2sinXcosX=sin2X. – Fernando Martinez Jul 28 '12 at 16:53
Of course. Let $X = \frac{x}{2}$. Also, you need to learn mathjax/latex. – copper.hat Jul 28 '12 at 16:54
I would but its kind of confusing to understand for me but I will try it. – Fernando Martinez Jul 28 '12 at 16:55
The formula $\sin x = 2 (\sin \frac{x}{2} \cos \frac{x}{2})$ holds no matter what you replace the symbol $x$ with. Replace it by, for example, $2Y$, then you get back the formula: $\sin (2Y) = 2 \sin Y \cos Y$. Now replace $Y$ by $X$ or $x$ if you want. – copper.hat Jul 28 '12 at 16:57
Oh no I was talking about latex I understand your first comment. – Fernando Martinez Jul 28 '12 at 16:58