Answer
See the detailed answer below.
Work Step by Step
We have here an equation of the induced surface charge density
on the dielectric in a capacitor,
$$\eta_{\rm induced}=\eta_0\left[1-\dfrac{1}{\kappa} \right]$$
We know that the polarized dielectric material forms another capacitor inside the original one, as we see in Figure 29.32 in your textbook.
So the net electric field inside the capacitor as one unit is given by
$$E=E_0-E_{\rm induced}$$
Hence,
$$E_{\rm induced}=E_0-E$$
Multiplying both sides by $\epsilon_0$ since we know that $\eta= \epsilon_0 E$;
$$\epsilon_0 E_{\rm induced}=\epsilon_0 E_0-\epsilon_0 E$$
$$\eta_{\rm induced}=\epsilon_0 E_0-\epsilon_0 E$$
$$\eta_{\rm induced}=\epsilon_0 (E_0- E)$$
$$\eta_{\rm induced}= \epsilon_0 E_0 \left(1- \dfrac{E}{E_0}\right)$$
Recalling that $\kappa=E_0/E$, so that $E/E_0=1/\kappa$
$$\boxed{\eta_{\rm induced}= \eta_0 \left(1- \dfrac{1}{\kappa}\right)}$$
