Answer
$\beta = arctan(\frac{(x+y)~tan~\alpha}{x})$
Work Step by Step
From the diagram:
$tan~\alpha = \frac{x}{z}$
$z = \frac{x}{tan~\alpha}$
$tan~\beta = \frac{x+y}{z}$
$z = \frac{x+y}{tan~\beta}$
We can equate the two expressions for $z$:
$\frac{x}{tan~\alpha} = \frac{x+y}{tan~\beta}$
$\frac{tan~\alpha}{tan~\beta} = \frac{x}{x+y}$
We can solve this equation for $\alpha$:
$\frac{tan~\alpha}{tan~\beta} = \frac{x}{x+y}$
$tan~\beta = \frac{(x+y)~tan~\alpha}{x}$
$\beta = arctan(\frac{(x+y)~tan~\alpha}{x})$
