We study the quasi-periodic Schrödinger operator\[(H_xu)_n = -u_{n+1} -u_{n-1} + \lambda V_0(x+n\omega)u_n\]on $l^2(\Z)$, where $V_0$ is a twice continuously differentiable, 1-periodic Morse function with exactly two critical points, $\omega$ is Diophantine, and $\lambda \gg 1$. Through the dynamics of the associated Schrödinger cocycle we reformulate spectral questions in terms of uniform hyperbolicity. The structure of the spectrum is then analyzed via the rotation number and the gap labeling theorem, which identifies the possible locations of spectral gaps. The main result shows that these gaps arise from resonances and provides explicit estimates for their locations and widths. Finally, we contrast the Diophantine setting with Liouville frequencies, where the discrete potential is well approximated by periodic sequences, and use this to establish the absence of point spectrum.