1. Relaciones de ángulos:
De $\alpha + \beta = \frac{\pi}{2} \implies \beta = \frac{\pi}{2} - \alpha$.
De $\alpha = \beta + \gamma \implies \gamma = \alpha - \beta$.

2. Uso de identidades:
Sustituimos $\beta$: $\gamma = \alpha - (\frac{\pi}{2} - \alpha) = 2\alpha - \frac{\pi}{2}$.
Esto es equivalente a $\frac{\pi}{2} + \gamma = 2\alpha$.
Tomamos tangente en $\gamma = \alpha - \beta$:
$$ \tan \gamma = \tan(\alpha - \beta) = \frac{\tan \alpha - \tan \beta}{1 + \tan \alpha \tan \beta} $$
Como $\alpha + \beta = \pi/2$, entonces $\tan \beta = \cot \alpha = 1/\tan \alpha$. Por lo tanto, $\tan \alpha \tan \beta = 1$.
$$ \tan \gamma = \frac{\tan \alpha - \tan \beta}{1 + 1} = \frac{\tan \alpha - \tan \beta}{2} $$
$$ 2 \tan \gamma = \tan \alpha - \tan \beta \implies \tan \alpha = 2 \tan \gamma + \tan \beta $$

$$ \boxed{\tan \beta + 2 \tan \gamma} $$