If $\tan(2x)=\frac{3}{4}$ and domain of $x =\left(0,\frac{\pi}{4}\right)$ find the value of $\tan(x)$

Question

How'd you approach this problem cause I was thinking about using an identity but I don't know how it'd be beneficial...

Thanks

Answer 1

Start with identity

$${3\over 4}=\tan{2x}={2\tan{x}\over 1-\tan^2{x}}$$

This quadratic in $u=\tan{x}$ has two solutions

$$\tan{x}=-3$$ $$\tan{x}={1\over 3}$$

But $x\in [0,\pi/4]$ so we retain the second solution

$$x=\arctan{1\over 3}$$