The sphere-to-plane projection of 360-degree video introduces substantial stretched redundant data, which is discarded when reprojected to the 3D sphere for display. Consequently, encoding and transmitting such redundant data is unnecessary. Highly redundant blocks can be referred to as all-zero blocks (AZBs). Detecting these AZBs in advance can reduce computational and transmission resource consumption. However, this cannot be achieved by existing AZB detection techniques due to the unawareness of the stretching redundancy. In this paper, we first derive a latitude-adaptive redundancy detection (LARD) approach to adaptively detect coefficients carrying redundancy in transformed blocks by modeling the dependency between valid frequency range and the stretching degree based on spectrum analysis. Utilizing LARD, a latitude-redundancy-aware AZB detection scheme tailored for fast 360-degree video coding (LRAS) is proposed to accelerate the encoding process. LRAS consists of three sequential stages: latitude-adaptive AZB (L-AZB) detection, latitude-adaptive genuine-AZB (LG-AZB) detection and latitude-adaptive pseudo-AZB (LP-AZB) detection. Specifically, L-AZB refers to the AZB introduced by projection. LARD is used to detect L-AZB directly. LG-AZB refers to the AZB after hard-decision quantization and zeroing redundant coefficients. A novel latitude-adaptive sum of absolute difference estimation model is built to derive the threshold for LG-AZB detection. LP-AZB refers to the AZB in terms of rate-distortion optimization considering redundancy. A latitude-adaptive rate-distortion model is established for LP-AZB detection. Experimental results show that LRAS can achieve an average total encoding time reduction of 25.85% and 20.38% under low-delay and random access configurations compared to the original HEVC encoder, with only 0.16% and 0.13% BDBR increases and 0.01dB BDPSNR loss, respectively. The transform and quantization time savings are 60.13% and 59.94% on average.