I understand how to find the stability of fixed points by using $f'(x)$ and inserting the values of the fixed points into that. However, I can't figure out how to tell the stability in systems such as
$$\frac{dx}{dt} = a + \sin(x) + \cos(2x)$$
In situations like this, the value of the fixed points varies as '$a$' changes, which I have no trouble finding, its just defining their stability that I struggle with.
Let us assume a system of this type
$$\dot{x}(t)=f(x;a)$$
with the equilibrium point $x_\text{eq}$. In this $a$ is a parameter. In this kind of problem, the equilibrium point is a function of the parameter $a$ (called bifurcation parameter in this context).
In order to obtain the equilibrium point(s) it is necessary to solve the nonlinear equation
$$0 = f(x_\text{eq};a)$$ $$\text{here: } 0 = a+\sin x_\text{eq}+1-2\sin^2x_\text{eq}$$
Then the stability of the equilibrium point can be obtained by Lyapunov's indirect method (also known as linearization).
Hence, you will have to determine
$$\left.\dfrac{\partial f}{\partial x}\right|_{x=x_\text{eq}}.$$
If this value is positive then, the equilibrium point is unstable. If it is negative hen the equilibrium point is at least locally asymptotically stable.