Trigonometric identity. How can I prove that $\sin A = 2\sin\frac{A}2\cos\frac{A}2$? How can I prove that $\sin A = 2\sin\frac{A}2\cos\frac{A}2$ ?
My failed take on this matter is:
$$
\sin A = 
\sin\frac{A}2 + \sin\frac{A}2 = 2\sin\Big(\frac{\frac{A}2+\frac{A}2}2\Big)\cos\Big(\frac{\frac{A}2-\frac{A}2}2\Big)=
2\sin\Big(\frac{\frac{2A}2}2\Big)\cos\frac02 =
2\sin\frac{A}2
$$
where $\cos\frac02 = \cos0 = 1$

Hint:
Your first step is wrong because
$\sin(x)$ is not a linear function so: $$\sin x \ne \sin(\frac{x}{2})+\sin(\frac{x}{2})$$
