How does cascading low pass modules affect pass band ripples?
The passband region of an equi-ripple filter is the region where its magnitude is in the interval $[1-\delta,1+\delta]$, where $\delta$ is the maximum approximation error. At the passband edge the value of the magnitude must equal $1-\delta$ (as shown in your drawing). The passband edge is the last frequency where the magnitude is still inside the interval $[1-\delta,1+\delta]$, and since the magnitude then falls off towards the stopband, the magnitude at the passband edge must be at the lower limit of the passband magnitude, which equals $1-\delta$.

The total magnitude response of a cascade of $M$ identical filters is

$$|H_{total}(\omega)|=|H(\omega)|^M$$

Consequently, the total magnitude at the passband edge is $(1-\delta)^M$. This value is then equal to $1-\delta_{total}$, which gives

$$\delta_{total}=1-(1-\delta)^M$$