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?

$$\alpha+\beta=\dfrac\pi2$$
$$\alpha-\beta=\gamma$$
Add to find $\alpha$ in terms of $\gamma$
Then apply $$\tan(x+y)$$ formula
