The textbook claims that this system of nonlinear equations is reversible. However I can't seem to work it out.
\begin{align} \dot{x} &= -2cos(x) - cos(y) \\ \dot{y} &= -2cos(y) - cos(x) \end{align}
By definition of reversible systems,
\begin{align} -\dot{x} &= -2cos(x) - cos(-y) = -2cos(x) - cos(y) \\ \dot{y} &= -2cos(-y) - cos(x) = -2cos(y) - cos(x) \end{align}
It seems like the reversibility holds for the 2nd equation but not for the first. Isn't the system known to be reversible if the system is invariant when t-> -t and y -> -y?
The system is reversible with respect to the transformation
$$t\rightarrow -t\\x\rightarrow -x\\y\rightarrow -y$$
as you can easily confirm. A transformation that changes only $y$ is not general. In the general sense, a set of ODE is called reversible if it is invariant under the transformation
$$t\rightarrow -t\\\boldsymbol{x}\rightarrow R\left(\boldsymbol{x}\right)$$
where $R^{2}\left(\boldsymbol{x}\right)=\boldsymbol{x}$.
You can check Strogatz's "Nonlinear Dynamics and Chaos" section $6.6$ on reversible systems. He gives there an example that is almost exactly identical to your ODE set (just replace $-2$ with $2$).