2 × tan 30° / $$(1 + tan^2 30°)$$ equals:
$$\dfrac{2\times tan 30^\circ}{1+tan^2 30^\circ}$$
$$= \dfrac{2\times\dfrac{1}{\sqrt{3}}}{1+(\dfrac{1}{\sqrt{3}})^2}$$
$$= \dfrac{\dfrac{2}{\sqrt{3}}}{\dfrac{4}{3}}$$
$$= \dfrac{2}{\sqrt{3}} \times \dfrac{3}{4}$$
$$ = \dfrac{3}{2\sqrt{3}} = \dfrac{\sqrt{3}}{2} = sin 60^\circ$$
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