Question

2 tan 30⁰ ÷ 1 - tan² 30⁰​

Answer 1

$tan(2x) = \frac{sin(2x)}{cos(2x)}\\tan(2x) = \frac{2*sin(x)*cos(x)}{cos^2(x)-sin^2(x)} \\\\tan(2x) = \frac{\frac{2*sin(x)*cos(x)}{cos^2x}}{\frac{cos^2x-sin^2x}{cos^2x} } \\\\tan(2x) = \frac{2*\frac{sin(x)}{cos(x)}}{1-\frac{sin^2(x)}{cos^2(x)} } \\tan(2x) = \frac{2tan(x)}{1-tan^2(x)}$

Thus,

$\frac{2tan(30)}{1-tan^2(30)} = tan(2*30) = tan(60) = \sqrt{3}$