We Have,

$\beta = \frac{I_c}{I_b} \hspace{1cm} \alpha = \frac{I_c}{I_e}$

$and I_e = I_c + I_b $

$\frac{I_c}{\alpha}=I_c + \frac{I_c}{\beta} \rightarrow \frac{1}{\alpha} = 1$

$\beta =\alpha\beta+\alpha = (\beta+1)\alpha$

$\alpha=\frac{\beta}{\beta+1}\hspace{1cm} \beta = \frac{\alpha}{\alpha-1}$