If two distinct tangents can be drawn from the point $(\alpha ,2)$ on different branches of the hyperbola $\frac{x^2}{9}-\frac{y^2}{16}=1\text{, }$, then

For two districts tangents on different branches, the point should lie on the line y = 2 and between A and B 

(where A and B are the points on the asymptotes).

 The equations of asymptotes are $4x=\pm 3y$

Solving with ! = 2, we have 

$\begin{array}{l}x=\pm \frac{3}{2}\\ \therefore -\frac{3}{2}<\alpha <\frac{3}{2}\end{array}$