Consider $$ u_t=u_{xx}+\cos u-1+\mu $$ and insert the traveling wave ansatz $u(x,t)=u(z), z=x-ct$. This gives $$ -cu_z=u_{zz}+\cos u-1+\mu $$ or, equivalently, the system \begin{align}\tag{1} &u_z=v\\ &v_z=-cv-\cos u+1-\mu. \end{align}
What is the linearization of this system in $u=u_0$?
I think one makes the ansatz $u=u_0+w$ and gets \begin{align} &u_z=w_z=v\\ &v_z=-cv-\cos(u_0+w)+1-\mu \end{align}
Now, how to linearize the term $\cos(u_0+w)$?
You have $\cos(u_0+w)=\cos(u_0) \cos(w)-\sin(u_0) \sin(w)$ so: $$\cos(u_0+w) = \cos(u_0) - \sin(u_0) w +O(|w|^2)$$ (Or even simpler $\cos$ is $C^1$ and $(\cos(u_0))'=-\sin(u_0)$).