How to prove a trigonometric identity $\tan(A)=\frac{\sin2A}{1+\cos 2A}$ Show that
$$
\tan(A)=\frac{\sin2A}{1+\cos 2A}
$$
I've tried a few methods, and it stumped my teacher.

The given equality is false. Set $A = \pi/2$. (Note: this applied to an earlier version of the problem).
Perhaps what you meant was
$$ \tan \frac{A}{2} = \frac{\sin A}{1 + \cos A}$$
or
$$ \tan A = \frac{\sin 2A}{1 + \cos 2A}$$
which is true, by using the half/double angle formulas.
$$\frac{\sin A}{1 + \cos A} = \frac{ 2 \sin A/2 \cos A/2}{2 \cos^2 A/2} = \tan A/2$$
