Half tangent representation

Question

If $x$ is defined by the interval $\pi/2>x>0$, and $\tan(x)=A$, what is $\tan(x/2)$? This is a multiple choice question on a test, and I don't have a approach because all the answer choices are in terms of A (the half angle formula for tangent involves cosine; not A). Any help would be appreciated.

Answer 1

Hint. Use $$\tan(2\theta)=\frac{2\tan\theta}{1-\tan^2\theta}$$ and let $\theta=x/2$.

Answer 2

Hint

You could start using the formula $$\tan(a+b)=\frac {\tan(a)+\tan(b)}{1-\tan(a)\tan(b)}$$ So, setting $b=a$ $$\tan(2a)=\frac {2\tan(a)}{1-\tan^2(a)}$$ dividing angles by $2$ then leads to $$\tan(a)=\frac {2\tan(\frac{a}{2})}{1-\tan^2(\frac{a}{2})}$$ This gives you a quadratic equation for $\tan(\frac{a}{2})$ the solutions of which being $$\tan(\frac{a}{2})=\frac{-1 \pm \sqrt{\tan(a)^2+1}}{\tan(a)}$$ which can be simplified in different ways. For sure, if $0 \leq a \leq \frac{\pi}{2}$, the $+$ sign should be used.