Question

Prove that $\tan (2\times {30}^{\circ })=\frac{2\tan {30}^{\circ }}{1-{\tan }^2{30}^{\circ }}$

Answer 1

LHS$=\tan (2\times 30)=\tan 60=\sqrt{3}$

RHS$=\frac{2\tan 30}{1-{\tan }^{2}30}=\frac{2\times \frac{1}{\sqrt{3}}}{1-\frac{1}{3}}=\frac{\frac{2}{\sqrt{3}}}{\frac{2}{3}}=\frac{\sqrt{3}}{\sqrt{3}}=\sqrt{3}$

RHS$=\sqrt{3}$

$\therefore$RHS=LHS

$\therefore \tan \left(60\right)=\frac{2\tan 30}{1-{\tan }^{2}30}$