This is known as the factor formula. It is used for the addition of sin functions. I don't understand how the two are equal though. How would you get to the right side of the equation using the left?
$$\sin(s) + \sin(t) = 2 \sin\left(\frac{s+t}{2}\right) \cos \left(\frac{s-t}{2}\right)$$
Well, notice you can choose real numbers $x$ and $y$ such that $$x=\frac{s+t}{2}, y=\frac{s-t}{2}$$ This gives $x+y=s$ and $x-y=t$, rewriting to $$\sin(x+y)+\sin(x-y)=2\sin(x)\cos(y)$$ Using the addition identity of the $\sin(x \pm y)=\sin(x)\cos(y) \pm\cos(x)\sin(y)$, this shows to be true.
It is derived from the following formula, a direct consequence of the addition formulæ: $$\sin (a+b)+\sin(a-b)=2\sin a\cos b?$$ Set $a+b=s$, $a-b=t$, and solve for $a$ and $b$.
