<s>[INST] If $A + B + C = \pi$, then show that $\sin(A) + \sin(B) + \sin(C) = 4\cos\frac{A}{2}\cos\frac{B}{2}\cos\frac{C}{2}$ So i have 
$A + B + C = \pi$
$$\frac{A}{2} + \frac {B}{2} + \frac{C}{2} = \frac{\pi}{2}$$
$$4\cos\left(\frac{-B-C + \pi}{2}\right)\cos\left(\frac{-A -C + \pi}{2}\right)\cdots$$
 And I doubt this leads to anywhere.
So then I tried, 
$\sin\left(\frac{-B-C + \pi}{2}\right)\cdots$ and this didn't go anywhere either. 
I don't know what to try, and I've seen other people's solutions and they do something like:
$\sin(C) = \sin(A + B)$, $\cos(C/2) = \sin(\frac{A + B}{2})$ but i don't see where they got this part from. Other people use Euler's formula or whatever but I haven't learned that yet so I can't use it.
 [/INST] $$\sin(A) + \sin(B) + \sin(C)= 2\sin\left(\frac{A+B}{2}\right)\cos\left(\frac{A - B}{2}\right) + 2\sin\left(\frac{C}{2}\right)\cos\left(\frac{C}{2}\right) $$
$$\sin(A) + \sin(B) + \sin(C)= 2\cos\left(\frac{C}{2}\right)\cos\left(\frac{A - B}{2}\right) + 2\sin\left(\frac{C}{2}\right)\cos\left(\frac{C}{2}\right)$$
$$\sin(A) + \sin(B) + \sin(C)= 2\cos\left(\frac{C}{2}\right)\cos\left(\frac{A - B}{2}\right) + \cos\left(\frac{A + B}{2}\right)$$
$$\sin(A) + \sin(B) + \sin(C)= 4\cos\left(\frac{A}{2}\right)\cos\left(\frac{B}{2}\right)\cos\left(\frac{C}{2}\right)$$
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