To use trigonometric identities to find the value of $\tan\alpha$ I'm stuck on this problem:  
If $\alpha+\beta=\frac{\pi}{2}$ and $\beta+\gamma=\alpha $ then find the value of $\tan\alpha$
I have tried isolating $\alpha$ but ended up getting $\tan\alpha = \frac{1}{\tan\beta}$ and $\tan\beta=\frac{1 - \tan\frac{\gamma}{2}}{1 + \tan\frac{\gamma}{2}}$   
Could someone please tell me how to properly approach this question?

Note that in terms of $\tan \beta$
$$\alpha+\beta=\frac{\pi}{2} \Rightarrow \alpha=\frac{\pi}{2}-\beta\implies\tan \alpha=\frac{1}{\tan \beta}$$
and in terms of $\tan \gamma$
$$\beta+\gamma=\alpha\implies\beta=\alpha-\gamma$$
$$\alpha=\frac{\pi}{2}-\alpha+\gamma\implies2\alpha=\frac{\pi}{2}+\gamma\implies\tan 2\alpha=\frac{2\tan \alpha}{1-\tan^2 \alpha}=-\frac{1}{\tan \gamma}\quad \tan\gamma\neq0\\\implies\tan \alpha=\tan\gamma\pm{\sqrt{\tan^2\gamma+1}}$$
