Answer
See proof below.
Work Step by Step
The Addition Formula for Tangent states that $tan(x+y)=\frac{tan(x)+tan(y)}{1-tan(x)tan(y)}$.
Let $y=x$ we get $tan(x+x)=\frac{tan(x)+tan(x)}{1-tan(x)tan(x)}$ Thus $tan(2x)=\frac{2tan(x)}{1-tan^2(x)}$
which is the Double-Angle Formula for Tangent.