Question

In a right triangle ABC,right angled at B,the ratio of AB to AC is 1:root 2.Find the value of 2tanA/1-tan square A

Answer 1

Since we know that ratio of AB : AC is 1 : √2, then we can more or less use those values for the sides, check the picture below.

so we're really looking for tan(2A).  Let's bear in mind that tan²(θ) is just another way to write [ tan(θ) ]².

$\bf \stackrel{\textit{Double Angle Identities}}{tan(2\theta)=\cfrac{2tan(\theta)}{1-tan^2(\theta)}}\\\\ -------------------------------\\\\ tan(2A)=\cfrac{2tan(A)}{1-tan^2(A)}\implies \cfrac{2\left( \sqrt{2} \right)}{1-(\sqrt{2})^2}\implies \cfrac{2\sqrt{2}}{1-2}\implies -2\sqrt{2}$