Feedback-linearization of a pendulum around $\delta_1\neq 0$

Question

In Non Linear Control Theory (Khalil) it is proposed the stabilization of the pendulum equation by feedback linearization.
In particular let $\delta_1\neq 0$ we want by $u=u_{SS}+u_\delta$ to stabilize the pendulum around this equilibrium point.
First of all it is opportune to find out $u_{SS}$ such that the system with this input has at the origin an equilibrium point and then we will design $u_\delta$ in order to stabilize the transformed system around the origin. So:
if the pendulum equation is given by: $\begin{cases} \dot x_1=x_2\\ \dot x_2=-bx_2-\sin{x_1}+cu \end{cases}$
I have thought (and it is confirmed by the book) that it is necessary a transformation like this:
$\begin{cases} z_1=x_1-\delta_1\\ z_2=x_2 \end{cases}$
from which then the resulting dynamic is:
$\begin{cases} \dot z_1=z_2\\ \dot z_2=-bz_2-\sin{(z_1+\delta_1)}+cu \end{cases}$
In order to obtain at the origin an equilibrium point I should have $0=-\sin{\delta_1}+cu$ and so $u_{SS}=\frac{\sin{\delta_1}}{c}$.
In another example of the same pendulum equation it is said:
a linearizing-stabilizing feeedback control is given by: $u=\frac{\sin{(z_1+\delta_1)}}{c}+u_\delta$.
So my question is: which one of the two cited $u_{SS}$ is correct? ($u=\frac{\sin{(z_1+\delta_1)}}{c}$ or $u=\frac{\sin{\delta_1}}{c}$?)