If α+β=(π)/(4) and 2 α=1 , then what is 2 β equal to?

$$\begin{aligned} & \because \quad 2 \tan \alpha=1 \quad \Rightarrow 2 \tan \alpha=\frac{1}{2} \\ & \alpha+\beta=\frac{\pi}{4} \quad \Rightarrow \quad \tan (\alpha+\beta)=\tan \frac{\pi}{4}\end{aligned}$$\Rightarrow \frac{\tan \alpha+\tan \beta}{1-\tan \alpha \cdot \tan \beta}=1\Rightarrow \tan \alpha+\tan \beta=1-\tan \alpha \cdot \tan \beta$$\begin{aligned} \Rightarrow & \frac{1}{2}+\tan \beta=1-\frac{1}{2} \cdot \tan \beta \\ \Rightarrow & 1+2 \tan \beta=2-\tan \beta \\ & \tan \beta=\frac{1}{3}\end{aligned}$$$Now,$\tan 2 \beta=\frac{2 \tan \beta}{1-\tan ^2 \beta}=\frac{2 \times \frac{1}{3}}{1-\frac{1}{9}}=\frac{2}{3} \times \frac{9}{8}=\frac{3}{4}$