Prove that :

$\text{tan} (2 \times 30°) = \dfrac{\text{2 tan 30°}}{1 - \text{tan}^2 \text{ 30°}}$

$\text{tan} (2 \times 30°) = \dfrac{\text{2 tan 30°}}{1 - \text{tan}^2 \text{ 30°}}$

L.H.S. = tan 2 x 30° = tan 60° = $\sqrt3$

R.H.S.

$= \dfrac{\text{2 tan 30°}}{1 - \text{tan}^2 \text{ 30°}}\\[1em] = \dfrac{2 \times \dfrac{1}{\sqrt3}}{1 - \Big(\dfrac{1}{\sqrt3}\Big)^2}\\[1em] = \dfrac{\dfrac{2}{\sqrt3}}{1 - \dfrac{1}{3}}\\[1em] = \dfrac{\dfrac{2}{\sqrt3}}{\dfrac{3}{3} - \dfrac{1}{3}}\\[1em] = \dfrac{\dfrac{2}{\sqrt3}}{\dfrac{3 - 1}{3}}\\[1em] = \dfrac{\dfrac{2}{\sqrt3}}{\dfrac{2}{3}}\\[1em] = \dfrac{2 \times 3}{2 \times \sqrt3}\\[1em] = \sqrt3$

∴ L.H.S. = R.H.S.

Hence, $\text{tan} (2 \times 30°) = \dfrac{\text{2 tan 30°}}{1 - \text{tan}^2 \text{ 30°}}$