# How do you prove that sin(2A) = (2tan(A))/(1 + tan^2(A))?

$\left(1\right) = 2 \sin A \cos A$
$\left(2\right) = \frac{2 \tan A}{1 + {\tan}^{2} A} = 2 \cdot \frac{\sin \frac{A}{\cos} A}{1 + {\sin}^{2} \frac{A}{\cos} ^ 2 a} = 2 \cdot \frac{\sin \frac{A}{\cos} A}{\frac{{\cos}^{2} A + {\sin}^{2} A}{\cos} ^ 2 A} =$
$= 2 \cdot \sin \frac{A}{\cos} A \cdot {\cos}^{2} \frac{A}{1} = 2 \sin A \cos A = \left(1\right)$.