The inequality is equivalent to
$\frac{\tan\beta}{\beta} >\frac{\tan\alpha}{\alpha}.$
This says that $(\tan x)/x$ is an increasing function on $(0,\pi/2)$. Differentiate:
$\frac{d}{dx}\frac{\tan x}{x}=\frac{2x-\sin2x}{2}\left(\frac{\sec x}{x}\right)^2.$
For $(\tan x)/x$ to be increasing we want to check that the above is be positive on the interval, which after dividing by obviously positive terms we deduce is equivalent to $u>\sin u$ on $u\in(0,\pi)$. We can deduce this by noting that both u'=1=\sin'(u) at $u=0$, and u'=1>\sin'u=\cos u on $[0,\pi)$. This implies $u>\sin u$ because, in general, $f\ge g$ on an interval entails $\int_a^u f(t)dt\ge \int_a^u g(t)dt$.