BALTIC-WAY 2005 Problem 2

Let $\alpha, \beta$ and $\gamma$ be three angles with $0 \leq \alpha, \beta, \gamma<90^{\circ}$ and $\sin \alpha+\sin \beta+\sin \gamma=1$. Show that

$$\tan ^{2} \alpha+\tan ^{2} \beta+\tan ^{2} \gamma \geq \frac{3}{8}$$