Question

prove that:

tan (2 x 30°) = ${\displaystyle \frac{2\tan 30°}{1–{\tan }^{2}30°}}$

Answer 1

RHS,

${\displaystyle \frac{2{\tan }^{2}30°}{1–{\tan }^{2}30°}=\frac{2\frac{1}{\sqrt{3}}}{1–\frac{1}{3}}=\frac{\frac{2}{\sqrt{3}}}{\frac{2}{3}}}$

= ${\displaystyle \frac{2}{\sqrt{3}}÷\frac{2}{3}}$

= ${\displaystyle \frac{\cancel{2}}{\sqrt{3}}\times \frac{3}{\cancel{2}}}$

= ${\displaystyle \frac{3}{\sqrt{3}}=\frac{\sqrt{3}\times \left(\cancel{\sqrt{3}}\right)}{\cancel{\sqrt{3}}}}$

= ${\displaystyle \sqrt{3}}$

LHS,
tan (2 x 30°) = tan 60° = ${\displaystyle \sqrt{3}}$
LHS = RHS