In this paper, we propose a faster-than-Nyquist (FTN) non-orthogonal frequency-division multiplexing (NOFDM) scheme for visible light communications (VLC) where the multiplexing/demultiplexing employs the inverse fractional cosine transform (IFrCT)/FrCT. Different to the common fractional Fourier transform-based NOFDM (FrFT-NOFDM) signal, FrCT-based NOFDM (FrCT-NOFDM) signal is real-valued which can be directly applied to the VLC systems without the expensive upconversion. Thus, FrCT-NOFDM is more suitable for the cost-sensitive VLC systems. Meanwhile, under the same transmission rate, FrCT-NOFDM signal occupies smaller bandwidth compared to OFDM signal. When the bandwidth compression factor $α$ is set to $0.8$, $20\%$ bandwidth saving can be obtained. Therefore, FrCT-NOFDM has higher spectral efficiency and suffers less high-frequency distortion compared to OFDM, which benefits the bandwidth-limited VLC systems. As the simulation results show, bit error rate (BER) performance of FrCT-NOFDM with $α$ of $0.9$ or $0.8$ is better than that of OFDM. Moreover, FrCT-NOFDM has a superior security performance. In conclusion, FrCT-NOFDM shows great potential for application in the future VLC systems.