The first part of the inequality can be rewritten as $$\frac{\cot\beta}{\cot\alpha} < \frac{\cos\beta}{\cos\alpha}\quad\Leftrightarrow\quad\frac{\cot\beta}{\cos\beta} < \frac{\cot\alpha}{\cos\alpha}\quad\Leftrightarrow\quad\sin\alpha <\sin\beta$$ This obviously holds because sine function is increasing in the region of $[0, \frac{\pi}{2}]$ and $0 < \alpha < \beta <\frac{\pi}{2}$. The second part of the inequality can be rewritten as $$\frac{\cos\beta}{\cos\alpha}<\frac{\beta}{\alpha}\quad\Leftrightarrow\quad\frac{\cos\beta}{\beta} <\frac{\cos\alpha}{\alpha}$$ This form is essentially to compare the slopes of two lines. To see this, let's take two points on the function $y=\cos{x}$: $A (\alpha, \cos\alpha)$ and $B (\beta,\cos\beta)$.

Because cosine function is decreasing in $[0, \frac{\pi}{2}]$ and $0 < \alpha < \beta < \frac{\pi}{2}$, therefore point $A$ is below the point $B$. Consequently, the slope of line $OA$ is smaller than $OAB$: $$k_{OA} < k_{OB} \implies \frac{\cos\alpha}{\alpha} < \frac{\cos\beta}{\beta}$$