In this study, we investigate the impact of zenith angle variations on cosmic-ray induced background in High-Purity Germanium (HPGe) gamma spectrometers using a coincidence technique based on plastic scintillator-Germanium detectors. We utilize an HPGe detector (Model GC2018 Mirion Ge Detector) enclosed within a low-activity cylindrical lead shield (Model 747E Mirion Lead Shield). For cosmic ray detection, a coincidence detection system with plastic scintillator detectors was positioned on top of the lead shielding. The zenith angle at the Germanium detector is computed using the dimensions of the square plastic scintillator and its distance from the Germanium detector center. We carried out measurements of cosmic-ray induced background in an HPGe gamma spectrometer with a square plastic configuration (80cm x 80cm), equivalent to a 45° zenith angle. The experimental measurements were compared with GEANT4 simulation data. The results demonstrate a good agreement between the measured energy spectrum and the simulated data across the energy range of 0.05 to 47 MeV. Further investigations into the effects of varying zenith angles provide valuable insights for optimizing HPGe spectrometer setups with minimized background interference.